hese Probability NOES elong to: Date opic Notes Questions. Intro 2. And & Or 3. Dependant & Independent. Dependant & Independent 5. Conditional 6. Conditional 7. Combinations & Permutations 8. inomial 9. Review 0. Review. ES Record any questions that you find challenging. ow to use this booklet. Do not do questions sequentially. Once you have got a concept, move on to the next set of questions. Once you have come to the end of a section, go back and complete the missing questions.
Memorize all definitions and terms: Probability Page 3 If an experiment has n equally likely outcomes of which r outcomes are r favorable to event A, then the probability of event A is: P ( A). n r option P(Rolling a ) n 6 options Probabilities must be between zero and one. heoretical Probability Page 3 Experimental Probability Page 3 Sample Space Page 3 0 P Page 3 AND Page 3 OR Page 3 P (a) vs p (a) Page,7 Complement Page heoretical Probability is what should happen. hese Probabilities are calculated using formulae. Experimental Probability is what actually happens. hese probabilities are calculated by experiment. he sample space is the set or the list of all possible outcomes. r he sample space is n in P ( A) n Probabilities are always between 0 and Probabilites can be written in decimal form or as fraction Probabilities can be converted to percentages. 0.2 20% (20% is not a probability it is a percentage) Means both hat is the probability that a randomly chosen student is a boy and in grade 2. Another way of saing this would be what is the probability that a Grade 2 boy is chosen. Multiply probabilties Means Either hat is the probability that a randomly chosen student is a boy or is in grade 2. Another way of saying this would be, what is the probability that the student chosen is a male student or a grade 2 girl. Add probabilities P (a) Probability that event a happens p(a) Probability that event a does not happen P (a) + p (a) he complement of a is a. If the probability of winning is 0.8, then the complement of winning is 0.2. he complement of winning is losing. P ( winning ) 0. 8 & P ( winning ) P ( Notinning) 0.2 Using the complement is often helpful when solving at least or at most questions. 2
Conditional probability Mutually exclusive Page 7 Independent events Page 6 Dependent events Page inomial Events & inomial Distribution Page 3 P ( A ) his is read, the probability of A given has already happened. he probability of A occurring under P(A ) will not be out of 0 but out of 5, the number of outcomes that are. Events are mutually exclusive if they have nothing in common Playing card example. earts and diamonds are mutually exclusive. Playing card example. earts and face cards are not mutually exclusive. Event A and are independent events if they do not impact each other. Examples of indepedent events: Rolling dice, Flipping coins, drawing cards with replacement Probabilities do not change if the events are independent Event A and are dependent if they impact each other in any way Example of dependent events: Drawing cards without replacement An event/distribution where there are only 2 options. Example of binomial events. Flipping a coin, Rolling a die and comparing rolling the number vs everything else. A binomial event always has a yes/no answer. Ie was the result a head? as the result a. as the result something other than? n C r (A) r (A NO) n-r 3
Introduction to Probability Probability If an experiment has n equally likely outcomes of which r outcomes are r favorable to event A, then the probability of event A is: P ( A). n r option P(Rolling a ) n 6 options Sample Space Page 3 0 P Page 3 OR And he sample space is the set or the list of all possible outcomes. r he sample space is n in P ( A) n Probabilities are always between 0 and Probabilites can be written in decimal form or as fraction Probabilities can be converted to percentages. 0.2 20% (20% is not a probability it is a percentage) Is inclusive. Is exclusive. A coin is flipped once. hat is the probability that the result will be heads? P(eads)0.5 or 2 Probabilities are always between zero and one. e can convert probabilities to percentages. here is a 50% chance that a coin will land heads. heoretical versus Experimental Probabilities A coin is tossed. Consider the event a single head.. Determine the heoretical probability 2. Determine the experimental probability with 00 tosses of the coin? heoretical probability is what should happen. Experimental probability is what actually happen. In Math 2 we will focus entirely on theoretical probabilities.
hat is the sample space for a regular deck of cards? Color Suit Non-Face Cards Face Cards Red earts A 2 3 5 6 7 8 9 0 J Q K Red Diamonds A 2 3 5 6 7 8 9 0 J Q K lack Clubs A 2 3 5 6 7 8 9 0 J Q K lack Spades A 2 3 5 6 7 8 9 0 J Q K Determine the probabilities. Leave your answer as a fraction in lowest terms. A single card is selected from a deck of 52 cards. hat is the probability of the following event? 3. P(eart). P(Red) 5. P(Club) 6. P(Red or lack) 7. P(Red & lack) 8. P(Club or face) 9. P(Club & Face) 0. P(lack 3). P(Not a heart) 2. P(Not a King) 3. P(eart or ). P(Jack) 5. P(Ace through 0) 6. P(Club or jack) Determine the sample space for each set of events. Use a tree diagram, a chart or a list to help. Leave your answer as a fraction in lowest terms. 7. A coin and a 6-sided dice are rolled. hat is the sample space? 22. wo sided dice are rolled. hat is the sample space? 27. hree coins are flipped. hat is the sample space? 8. P( and 6) 23. P( and 2) 28. P(At least head) 9. P( and odd) 2. P( or 2) 29. P(No heads) 20. P( or 6) 2. P( or odd) 25. P(Not a ) P ( ) 26. P(Sum is even) 30. -P(No heads) 5
Determine the probabilities and number of outcomes. A biased coin is weighted so that it lands heads 70% of the time. A biased coin weighted so that it lands tails 0% of the time. A biased die is weighted so that it returns a 3 20% of the time. 3. P() 35. P() 39. P(3) 32. P() 36. P() 0. P(6) 33. he biased coin is flipped 20 times. ow many heads will result? 37. he biased coin is flipped 80 times. ow many heads will result?. he biased 6-sided die is rolled 50 times. ow many 5s will result? 3. he biased coin is flipped 50 times. ow many tails will result? 38. he biased coin is flipped 60 times. ow many tails will result? 2. he biased 6-sided die is rolled 200 times. ow many even numbers will result? Calculate the following probabilities. A card is drawn from a shuffled deck of 52 cards. hat is the probability of each event? 3. hat is the sample space?. A red card is drawn. 5. A face card is drawn. 6. A heart is drawn. In the card game In etween, 3 cards from a deck of 52 are drawn. o win, the 3 rd card must be in between the first two cards. he player loses if the 3 rd card is the same as the first two. Determine the probability of winning given the first two cards already drawn. 7. hat is the sample space? 8. A 3 and a 7 are drawn. 9. A 5 and a Queen are drawn. 50. An 8 and a 9 are drawn. 6
Calculate the following probabilities. 2 six-sided dice are rolled. Determine the probability of each event Sample space 5. he sum is odd. 52. he sum is 6 or 2. 53. A double is rolled. 2 3 5 6 2 22 23 2 25 26 3 32 33 3 35 36 2 3 5 6 5 52 53 5 55 56 6 62 63 6 65 66 5. he sum of the 2 dice is at least 3. 55. he sum of the two dice is at most 0. Calculate the following probabilities. 56. If you guess on every question, what is the probability of getting 00% on a 3 question true false test? Solution: Determine the sample space first 57. If you guess on every question, what is the probability of getting 00% on a 5 question true-false test? 58. If you guess on every question, what is the probability of getting 00% on an 8 question true-false test? Question # 2 options Question #2 2 options Question #3 2 options 0.0325 0.0039 here are 8 possible answer 3 keys 2 2 2 2 8 59. If you guess on every question, what is the probability of getting 00% on a 5 question multiple choice test?(acd) 60. If you guess on every question, what is the probability of getting 00% on a 0 question multiple choice test?(acd) here is one correct answer key P(00%) 8 0.25 0.000977 0.000000953 7
OR, AND and Probability AND Means both hat is the probability that a randomly chosen student is a boy and in grade 2. Another way of saying this would be what is the probability that a Grade 2 boy is chosen. Multiply probabilties OR Means Either hat is the probability that a randomly chosen student is a boy or is in grade 2. Another way of saying this would be, what is the probability that the student chosen is a male student or a grade 2 girl. Add probabilities P (a) vs p (a) P (a) Probability that event a happens p(a) Probability that event a does not happen P (a) + p (a) he complement of a is a. Complement If the probability of winning is 0.8, then the complement of winning is 0.2. he complement of winning is losing. Conditional probability P ( A ) his is read, the probability of A given has already happened. he probability of A occurring under P(A ) will not be out of 0 but out of 5, the number of outcomes that are. Events A and are mutually exclusive. Events A and are not mutually exclusive. here is no overlap of A and P(A and )0 here is overlap of A and. P(A and )3/9 8
Calculate the probabilities. 6. P (A) 62. P () Venn Diagram 63. P (A) (Not A) 65. P ( A & ) oth A and 6. P () 66. P (Aor ) Either A or or both 67. p (Aor ) 68. P ( A & ) 69. P(Neither) 70. P(Only one) Only A or only 7. ( A ) P 72. P ( A) P ( A ) his is read, the probability of A given has already happened. he probability of A occurring under P(A ) will not be out of 0 but out of 5, the number of outcomes that are. Calculate the probabilities. P ( A & ) 7. P ( A ) 73. Venn Diagram 75. P(Neither) 76. P (Aor ) 77. P (A) 78. p (Aor ) 79. P ( A) 80. P(only one) 8. P () 82. P (A) 83. ( A & ) P 8. P () 85. ( or) P 86. P ( and ) 87. P ( Aor) 88. P ( Aand ) 9
Challenge #: A market study found that 50% of a neighborhood likes Japanese food while 60% likes Italian food. 30% like both. Determine the following probabilities: P(likes at least ) P(likes only one type of food) Challenge #2: A V station determined that 30% of boys watch sports and 60% watch soaps. 20% watch neither. Determine the following probabilities: P(oy watches at most show) P(oy watches at least show) Use Venn diagrams to calculate the following probabilities. A market study found that 50% of a neighborhood like Japanese food while 60% like Italian food. 30% like both. 89. P(likes at least ) A V station determined that 30% of boys watch sports and 60% watch soaps. 20% watch neither. 9. P(oy watches at most show) 6% of girls want to go into business and % want to go into education. % want neither. 90. P(likes only one type of food) 92. P(oy watches at least show) 93. P(Girl pursue at most one career) Solution: Draw and label a Venn diagram 30% goes in the overlap Fill out the rest of the diagram by subtracting 30% from 50% and 60% Solution: All percentages must add to 00%. Determine the overlap by adding all percentages and then subtracting by 00%. Use the diagram to calculate the above probabilities. Use the diagram to calculate the above probabilities. 0
A market Study found that 0% of a neighborhood likes Japanese food while 50% likes Italian food. 30% like both. 9. P(likes at least ) 95. P(likes only one type of food) A V station determined that 20% of boys watch sports and 50% watch the news. 0% watch neither. 96. P(oy watches only sports) 97. P(oy watches the news or nothing) 80% of girls want to go into business and 30% want to go into education. 20% want neither. 98. P(Girl pursues both careers) 99. P(Girl pursue at most one career) Challenge #3: A card is drawn from a shuffled deck of 52 cards. Determine the probability that the card is a heart or a spade. Challenge #: A card is drawn from a shuffled deck of 52 cards. Determine the probability that the card is a heart or a face card.
Calculate the following probabilities. Leave your answer as a fraction in lowest terms. A card is drawn from a shuffled deck of 52 cards. Determine the probability of each event. 00. he card is a heart or a spade. Solution: hese events are mutually exclusive: 0. he card is an ace or a face card. 02. he card is a 0 or a face card. 03. he card is a spade or a red. P()+P(S) 3 3 + 52 52 2 0. he card is a heart or a face card. 05. he card is an ace or a spade. 06. he card is a spade or a black 8. 07. he card is a club or a black card. Solution: hese events are not mutually exclusive P()+P(F)-P(&F) 3 52 + 2 52 3 52 22 52 26 Challenge #5: A survey of 200 people indicated that 60 learn from the newspaper, 50 from the V and 30 from both sources. Determine the following probabilities: P(A randomly selected person learns from the newspaper and V) P(A randomly selected person learns from at least one of the sources) P(A randomly selected person learns from exactly one of the sources) Challenge #6: A study of hand-eye coordination tested people on how quickly they could respond to a moving object on a screen. 30% of the people responded in less than 0.3s; 60% in 0.5s or less; and 5% took more than 0.8s. Determine the following probabilities: P(A randomly selected person from this group will take 0.8s or less) P(A randomly selected person from this group will take longer than 0.5) P(A randomly selected person from this group will take between 0.3 and 0.5 inclusive) 2
A survey of 200 people indicated that 60 learn from the newspaper, 50 from the V and 30 from both sources. hat is the probability of each event? Venn Diagram 08. P(A randomly selected person learns from the newspaper and V) 09. P(A randomly selected person learns from at least one of the sources) 0. P(A randomly selected person learns from exactly one of the sources) A study of hand-eye coordination tested people on how quickly they could respond to a moving object on a screen. 30% of the people responded in less than 0.3s; 60% in 0.5s or less; and 5% took more than 0.8s. hat is the probability of each event? Venn Diagram. P(A randomly selected person from this group will take 0.8s or less) 2. P(A randomly selected person from this group will take longer than 0.5) 3. P(A randomly selected person from this group will take between 0.3 and 0.5 inclusive) A study of hand-eye coordination tested people on how quickly they could respond to a moving object on a screen. 25% of the people responded in less than 0.3s; 55% in 0.5s or less; and 0% took more than 0.8s. hat is the probability of each event? Venn Diagram. P(A randomly selected person from this group will take 0.8s or less) 5. P(A randomly selected person from this group will take longer than 0.5) 6. P(A randomly selected person from this group will take between 0.3 and 0.5 inclusive) Challenge #7: In a recent survey of grade 2 students, it was found that 70% took math and 50% took chemistry. Determine the chance of each event. If 80% took math or chemistry, what percent of students took math only? 3
Calculate the percentage. Venn Diagram 7. In a recent survey of grade 2 students, it was found that 70% took math and 50% took chemistry. Determine the chance of each event. If 80% took math or chemistry, what percent of students took math only? 9. In a recent survey of grade 2 students, it was found that 70% took math and 50% took chemistry. Determine the chance of each event. If 80% took math or chemistry, what percent of students took both math and chemistry? 8. Chemistry only? 20. Neither? Venn Diagram 2. In a recent survey of grade 2 students, it was found that 65% took math and 5% took chemistry. If 30% took math and chemistry, what percent of students took chemistry only? 23. In a recent survey of grade 2 students, it was found that 65% took math and 5% took chemistry. If 30% took math and chemistry, what percent of students took neither math nor chemistry? 22. Math Only? 2. ook both? Challenge #8: Determine the probability of flipping three heads in three flips. Challenge #9: Determine the probability of flipping one head in three flips.
Dependent and Independent Probabilities Complement he complement of an event happening is the probability that it does not happen P ( winning ) 0. 8 & P ( winning ) P ( Notinning) 0. 2 Using the complement is often helpful when solving an at least or at most question. P (a) vs p (a) P (a) Probability that event a happens p(a) Probability that even a does not happen P (a) + p (a) OR P (a) - p (a) A coin is flipped 3 times. Determine the probabilities as fractions in lowest terms. ere are the possible outcomes: 25. hat is the probability of flipping 3 heads? 26. hat is the probability of flipping head in 3 flips? 27. hat is the probability of flipping 2 heads in 3 flips? 28. hat is the probability of flipping at least head? Challenge #0: immy has a bag full of marbles. here are 6 black marbles and white marbles in the bag. Determine the following probabilities if he removes 2 marbles from the bag without replacement. Determine the probability that both marbles are black. Challenge #: immy has a bag full of marbles. here are 6 black marbles and white marbles in the bag. e takes out one marble, looks at it, puts it back in the bag and then randomly draws another marble. Determine the probability that exactly one of marbles is black. 5
Dependent probabilities immy has a bag full of marbles. here are 6 black marbles and white marbles in the bag. Determine the following probabilities if he removes 2 marbles from the bag without replacement. ree diagram: 29. P(oth are black) 30. P(Exactly is black) 0.333 0.533 3. hat is the probability that at least marble will be black? 0.867 Independent probabilities immy has a bag full of marbles. here are 6 black marbles and white marbles in the bag. e takes out one marble, looks at it, puts it back in the bag and then randomly draws another marble. ree diagram: 32. P(oth are black) 33. P(Exactly is black) 3. hat is the probability that at least marble will be black? 0.36 0.8 0.8 6
Calculate the following probabilities. Round your answer to 3 decimals. immy has a bag full of marbles. here are 3 black marbles, white marbles and 3 striped marble in the bag. e removes 2 marbles one at a time without replacement. 35. P(oth are black) 36. P(Exactly is black) 0.067 0.67 37. P( & S) 38. P( & S) 0.2 0.267 immy has a bag full of marbles. here are 3 black marbles, white marbles and 3 striped marble in the bag. e takes out one marble, looks at it, puts it back and then randomly draws another marble. 39. P(oth are black) 0. P(Exactly is black) 0.09 0.2. P( & S) 2. P( & S) 0.8 0.2 7
2 cards are removed from a deck of 52 cards without replacement. Determine the following. 3. P( heart & spade). P( five & four) 5. P( face card & ace) Remember: P( & S) P( & S) or P(S & ) 0.27 0.02 0.036 6. P( ten & two) 7. P( red & black) 8. P( face card & six) 0.02 0.50 0.036 Calculate the following probabilities. Round your answer to 3 decimals. A biased coin with P(eads)0.7 was tossed 3 times. Determine the following. 9. ree diagram 50. P(,, in that order) 5. P(,, in that order) 0.063 0.7 52. P(,,) 53. P(&2) 0.027 0.89 5. P(2 &) 55. hat is the probability of flipping at least head? 0. 0.973 8
Challenge #3: he probability that you are late for class is 0.2. he probability that your teacher is late is 0.. Determine the following probabilities: P(both on time) P( of you is late) Challenge #: In 5 tosses of a coin, the first 2 tosses resulted in 2 heads. hat is the probability that the 5 tosses will produce exactly 3 heads? Calculate the following probabilities. he probability that you are late for class is 0.2. he probability that your teacher is late is 0...If these events are independent, determine the following probabilities. 56. ree Diagram 57. P(both on time) 58. P( of you is late) 0.72 0.26 he probability that a student completes her math is 0.6. he probability that she completes her French is 0.3. If these events are independent, determine the following probabilities. 59. ree Diagram 60. P(No done) 6. P(Only complete) 62. P(both complete) 0.28 0.5 0.8 9
Calculate the following probabilities. Round your answer to 3 decimals. 63. In 5 tosses of a coin, the first 2 tosses resulted in 2 heads. hat is the probability that the 5 tosses will produce exactly 3 heads? Solution: P(,2 2)hat is the probability that a third head will be flipped after 2 have already occurred. ecause two heads have already occurred, we will solve this as a 3 flip question rather than a 5 flip question P(, 2). 6. In 7 tosses of a coin, the first 5 tosses resulted in 3 tails and 2 heads. hat is the probability that the 7 tosses will produce exactly 3 heads? 65. In 80 tosses of a coin, the first 78 tosses resulted in 78 tails. hat is the probability that the 80 tosses will produce exactly 2 heads? 66. immy is planning to role a die 6 times. After he rolled 2 fives and 2 threes, he wonders what the probability that the 6 tosses will result in 3 fives and 3 threes. 0.5 0.25 67. immy is planning to role a die 6 times. After he rolled 2 fives and 2 threes, he wonders what the probability that the 6 tosses will result in 2 fives and threes. P(, 2 ) P( ) or P( ) or P( ) 2 2 2 + 2 2 2 + 2 2 2 3 8 0.056 0.0278 Challenge #5: he probability that a battery will last one month is 0.7 and that it will last 2 months is 0.2. At the end of the first month, what is the probability that the battery will also last until the end of the 2 nd month? Challenge #6: here is a 5% chance that a car will malfunction in the st year and a 0% chance in the first 2 years. hat is the probability that the car lasts the first year but breaks down in the 2 nd year? 20
68. he probability that a battery will last one month is 0.7 and that it will last 2 months is 0.2. At the end of the first month, what is the probability that the battery will also last until the end of the 2 nd month? 69. he probability that a battery will last one month is 0.8 and that it will last 3 months is 0.. At the end of the first month, what is the probability that the battery will also last until the end of the 3 rd month? 70. he probability that your new I- ook lasts 3 years is 0.9 and that it will last 5 years is 0.3. After 3 years, what is the chance that your I-ook will last another 2 years? P(Lasts 2 months)p(lasts st &2 nd ) P(L ) and P(L 2 L ) 0.2(0.7)(A) A0.2857 0.5 0.333 7. here is a 5% chance that a car will malfunction in the st year and a 0% chance in the first 2 years. hat is the probability that the car lasts the first year but breaks down in the 2 nd year? 72. he probability that your new I- ook last 3 years is 0.9 and that it will last 5 years is 0.3. After 3 years what is the chance that your I-ook will break down in the next two years? 73. he probability that a battery will last one month is 0.6 and that it will last 2 months is 0.3 At the end of the first month, what is the probability that the battery will also last until the end of the 2 nd month? P(lasts 2 years)p(lasts st ) & P(lasts 2 nd ) 0.9(0.95)(L 2) L 20.97 P(reaks in the 2 nd year)-p(lasts 2 nd ) 2-0.97 0.053 0.667 0.5 2
Determine the following probabilities. Round your answer to 3 decimals. If a fair six-sided die is tossed twice, determine the following probabilities: 7. Determine the probability that the first toss is greater than 2 and the second is less than 2. 75. Determine the probability that the first toss is greater than 5 and the second is less than 5. 76. Determine the probability that the first toss is greater than and the second is less than. 0. 0. 0.67 77. A bag contains 5 red balls and 0 black balls. wo balls are drawn from the bag, one after the other, without replacement. Determine the probability that one of each is chosen. 78. A bag contains 5 red balls and n black balls. wo balls are drawn from the bag, one after the other, without replacement. Determine an expression to represent the probability that one of each is chosen. 79. A bag contains x red balls and 0 black balls. wo balls are drawn from the bag, one after the other, without replacement. Determine an expression to represent the probability that one of each is chosen. 22
nc r, ree Diagrams and Probabilities. immy takes marbles out of the bag at the same time. Determine the probability for the following events: P(), P(3), P(2), P(), P(0). # OF LACK 3 3 2 3 2 2 3 2 2 2 0 # OF LACK ow many pathways to s? ow many pathways to 3 s? ow many pathways to 2 s? ow many pathways to? ow many pathways to 0 s? C C 3 C 2 6 C C 0 6 0 5 3 9 8 7 6 0 6 0 6 0 0 5 9 8 7 5 9 8 7 5 9 8 7 6 5 9 8 7 2 2 2 2 2 2 2 2 he probability of each pathway is the same regardless of the order of the fractions. 6 0 6 0 6 0 0 0 0 5 3 9 8 7 5 3 9 8 7 3 5 9 8 7 6 5 3 9 8 7 6 3 5 9 8 7 3 6 5 9 8 7 6 0 0 0 0 3 2 9 8 7 6 3 2 9 8 7 3 6 2 9 8 7 3 2 6 9 8 7 35 35 35 35 he probability of each pathway is the same regardless of order of the fractions. 0 3 2 9 8 7 20 6 5 3 C 0 9 8 7 6 5 C3 0 9 8 7 8 2 6 5 3 C 2 0 9 8 7 3 7 6 3 2 C 0 9 8 7 35 3 2 C0 0 9 8 7 20 Remember: he order of fractions does not matter. 23
Challenge #7: Little immy removed cards without replacement from a deck of 52 cards. Determine the probability of drawing exactly 3 out of kings. Challenge #8: A biased coin with P(eads)0.7 was tossed 5 times. Determine the probability that exactly out of the 5 flips will land heads. Calculate the following probabilities. Round your answer to decimals. Little immy removed cards without replacement from a deck of 52 cards. ree Diagram 80. P(3Kings) 8. P( Kings) 82. P(2 Kings) C 3( KKK,0) 3 2 8 C 3 52 5 50 9 0.0007 0.0000(037) 0.0260 Little immy removed cards without replacement from a deck of 52 cards. 83. hat is the probability of picking at least five? 8. omit 85. hat is the probability of picking at least heart? 0.997 0.696 2
Calculate the following probabilities. Round your answer to 3 decimals. A biased coin with P(eads)0.7 was tossed 5 times. Calculate each probability. ree Diagram: 86. P(5) 87. P() Solution: nc r () 5 flips choose heads 5C (0.7)(0.7)(0.7)(0.7)(0.3) 0.3605 88. P(3) 89. P() 0.68 0.360 0.32 0.028 90. hat is the probability of flipping at least 2 heads? 9. hat is the probability of flipping at least head? 0.969 0.998 25
Calculate the following probabilities. Round your answer to 3 decimals. 5 balls are removed without replacement from a bag containing 5 red balls and 6 yellow balls. 92. ree Diagram 93. P(3 Red) 9. P( red) 95. P(5 red) 96. hat is the probability of picking at least 3 red? 0.325 0.069 0.002 97. hat is the probability of picking at least red? 0.3977? 0.987 Challenge #9: wo cards are drawn without replacement from a shuffled deck of 52 cards. hat is the probaility that the second card will be an ace? Round your answer to 3 decimals. Challenge #20: wo cards are drawn without replacement from a shuffled deck of 52 cards. hat is the probaility that the first card is a heart and the second card is the king of hearts. 26
Conditional Probability wo cards are drawn without replacement from a shuffled deck of 52 cards. hat is the probaility that the second card will be the following card? Round your answer to 3 decimals. 98. An ace is the second card 99. A red five is the 2 nd card 200. he king of spades is the 2 nd card. Solution: his a conditional probability question because probabilities are conditional based on the first card chosen. here are 2 options for the first card, an ace or not an ace. 0.038 0.09 20. A face card is the 2 nd card. 202. he 0 of clubs is the second card P ( A 2 ) P ( A, A) orp ( A, A) 3 8 + 52 5 52 5 0.0769 52 3 0.23 0.09 wo cards are drawn without replacement from a shuffled deck of 52 cards. Determine the following probabilities. Round your answer to 3 decimals. 203. st eart, 2 nd King of earts 20. st Red, 2 nd 8 of earts 205. st face Card, 2 nd K of earts Solution: P( st K 2 nd ) 2 52 5 2 2652 he first card can not be the king of hearts. hat is why there are 2 options and not 3. 0.009 0.00 27
Color Suit Non-Face Cards Face Cards Red earts A 2 3 5 6 7 8 9 0 J Q K Red Diamonds A 2 3 5 6 7 8 9 0 J Q K lack Clubs A 2 3 5 6 7 8 9 0 J Q K lack Spades A 2 3 5 6 7 8 9 0 J Q K wo cards are drawn without replacement from a shuffled deck of 52 cards. hat is the probability that the second card is each card? Round your answer to 3 decimals. 206. A 0 is the 2 nd card. 207. A face card is the 2 nd card. 208. A red card is the 2 nd card. 0.077 0.23 0.5 209. An even number is the 2 nd card. 20. he 5 of clubs is the 2 nd card. 2. A diamond is the 2 nd card. 0.385 0.09 0.25 wo cards are drawn without replacement from a shuffled deck of 52 cards. Determine the following probabilities. Round your answer to 3 decimals. 22. st Face, 2 nd King 23. st Red, 2 nd Red Face 2. st Club, 2 nd 8 of Clubs 0.07 0.057 0.005 28
Understanding ayes Law Challenge #9: 00 grade 2 students were surveyed. 60 were boys and 0 were girls. 80% of the boys and 70% of the girls who were surveyed said they love chocolate ice cream. Examine the following two questions and calculate each probability below. Determine the probability that a randomly selected student likes chocolate ice cream.(25) A student is chosen who likes chocolate ice cream, determine the probability that selected student is a boy.(29) Can you see the difference between these two questions? 25. hat is the probability that a randomly selected math 2 student loves math 2? 26. ow many boys surveyed love chocolate ice cream? 27. ow many girls surveyed love chocolate ice cream? 28. ow many of the students surveyed love chocolate ice cream? 0.76 29. A student is chosen who likes chocolate ice cream, determine the probability that selected student is a boy. A student is randomly chosen and loves math. hat is the probability that it is a boy? Number of boys who love chocolate ice cream Number of students who love chocolate ice cream his is ayes Law 220. P(Student who love chocolate ice cream is a boy)0.6358 29
22. A computer supply store buys 0% of their computer chips from igchips and 60% from Fastchips. On average, 6% of the igchips are faulty and 5% of the Fastchips are faulty. If a randomly selected chip is faulty, what is the probability that igchips made it? 0. Calculate the following probabilities. Round your answer to 3 decimals. here are 00 boys and 20 girls in the grade 2 year. 20 boys and 30 girls have no siblings. A student is randomly selected. 222. ree Diagram 223. hat is the probability that the student has no siblings? 22. A student is chosen who has no siblings. hat is the probability that the student is a girl? 0.227 0.6 here are 00 boys and 50 girls in the grade 2 year. 0 boys and 0 girls have no siblings. A student is randomly selected. 225. ree Diagram 226. hat is the probability that the student has no siblings? 227. A student is chosen who has no siblings. hat is the probability that the student is a girl? 0.333 0.2 30
Calculate the following probabilities. A new medical test for tsprayitis is 95% accurate. Suppose 8% of the population have tsprayitis. hat is the probability of each event? 228. ree Diagram 229. A randomly selected person will test negative. 230. A person tests negative, calculate the probability that they have actually have tsprayitis. 23. A person tests positive, caculate the probability that they do not have tsprayitis. 0.878 0.005 0.377 A new medical test for bad hair is 90% accurate. Suppose 20% of the population has bad hair. hat is the probability of each event? 232. ree Diagram 233. A randomly selected person will test negative. 23. A person tests negative, calculate the probability that they actually have bad hair. 235. A person tests positive, calculate the probability that they actually have good hair. 0.7 0.027 0.308 A new medical test for Simonsonrea is 80% accurate. Suppose 30% of the population has Siminsonrea. hat is the probability of each event? 236. ree Diagram 237. A randomly selected person will test negative. 238. A person tests negative, caclulate the probablity that they do not have Simonsonrea. 239. A person tests positive, calculate the probability that they actually have Simonsonrea. 0.62 0.903 0.632 3
20. here are two bags full of marbles. One bag is black and one bag is green. he black bag contains 6 white balls and 2 red balls. he green bag contains 2 red balls and 6 white balls. A marble is randomly selected from the black bag and placed in the green bag. A marble is then randomly selected from the green bag. Challenge #2: hat is the probability that the ball selected from the green bag is white? Challenge #22: If a white ball is selected from the green bag, what is the probability that a red ball was transferred from the black bag to the green bag? Calculate the following probabilities. Little immy has a black bag and a green bag. he black bag contains 6 white balls and 2 red balls. he green bag contains 2 red balls and 6 white balls. A marble is randomly selected from the black bag and placed in the green bag. A marble is then randomly selected from the green bag. 2. hat is the probability that the ball selected from the green bag is white? 22. If a white ball is selected from the green bag, what is Solution: here are 2 options: oth balls are white 2 st is not white, but 2 nd is P(2 nd ball is white) the probability that a red ball was transferred from the black bag to the green bag? P( ) or P( R) 6 7 2 6 5 3 + 8 9 8 9 72 0.222 23. hat is the probability that the ball selected from the green bag is red? 2. If a red ball is selected from the green bag, what is the probability that a red ball was transferred from the black bag to the green bag? 25. If a white ball is selected from the green bag, what is the probability that a white ball was transferred from the black bag to the green bag? 0.25 0.333? 0.778 32
Little immy has lack ag and a Green bag. he black bag contains red ball and 6 white balls. he green bag contains 3 red balls and 2 white balls. 26. A marble is randomly selected from the black bag and placed in the green bag. A marble is then randomly selected from the green bag. hat is the probability that the ball selected from the green bag is white? 27. If a white ball is selected from the green bag, what is the probability that a white marble was transferred from the black bag to the green bag? 0.33 0.692 Calculate the following probabilities. Round your answer to 3 decimals. Little immy has black bag and a green bag. he black bag contains 7 red marbles and 3 white marbles. he green bag contains 3 red marbles and white marbles. 28. A marble is randomly selected from the black bag and placed in the green bag. A marble is then randomly selected from the green bag. hat is the probability that the marble selected from the green bag is white? 29. If a white marble is selected from the green bag, what is the probability that a white marble was transferred from the black bag to the green bag? 0.26 0.62 250. A black bag contains 6 red balls and green balls. A white bag contains red balls and green ball. A regular die is rolled and if a comes up, a ball is selected randomly from the black bag. Otherwise, a ball will be selected randomly from the white bag. If a green ball is selected, what is the probability that it came from the white bag? 0.7 33
Permutations, Combinations and Probabilities Color Suit Non-Face Cards Face Cards Red earts A 2 3 5 6 7 8 9 0 J Q K Red Diamonds A 2 3 5 6 7 8 9 0 J Q K lack Clubs A 2 3 5 6 7 8 9 0 J Q K lack Spades A 2 3 5 6 7 8 9 0 J Q K Calculate the number of possible 5 card hands. Find the number of 5 Formula card hands made up of: 25. five, seven, 3 other cards C x Cx 3 C 5 s choose, 7 s choose, cards that are not 5s or 7s choose 3 Number 290 252. Any cards 2598960 253. earts only 287 25. 2 fours, 3 fives 2 255. kings, ace Calculate the following probabilities. Round your answer to 3 decimals. Find the probability of the Formula Probability following 5 card hands: 256. 3 fives, 2 other cards C 3x 8C 2 C 257. 3 black,2 hearts 258. heart,3 clubs, 5 of spades 259. red, king of clubs 260. A,K,Q,J,0 26. 2 fours, 2 fives, any other card 262. Sevens, any other card 52 5 0.002 0.078 0.00 0.0057 0.000(000) 0.00 0.000(02)? 3
people are randomly selected from a group of 8 boys and 6 girls to represent the school in the debate championships. Challenge #20: ow many unique groups can be created from the group of students? Challenge #2: Determine the probability that exactly 3 of the four students chosen are girls. ow many unique groups can be created with exactly 3 girls? 00 60 Calculate the number of possibilities. Round your answer to 3 decimals. A bowling team is made up of 6 boys and girls. 263. ow many different groups of can be sent to the city championships if 2 boys and 2 girls have to go? 26. ow many different groups of can be sent to the city championships? 265. ow many different groups of can be sent to the city championships if boy and 3 girls have to go? Solution Choose 2 boys from 6 6 C 2 Choose 2 girls from C 2 Multiply the results C 6 2 C2 90 20 2 Calculate the following probabilities. people are to be randomly selected from a group of 8 boys and 6 girls. 266. P(Exactly 3 girls) 267. P(All boys) 268. P(Alternate in gender) 0.60 0.070 0.399 35
6 people are randomly selected from a group of 0 boys and 2 girls. 269. P(Exactly 5 boys) 270. P(All boys) 27. P(Alternate in gender) Calculate the following probabilities. 0.0 0.003 0.035 Of the 20 students on this year s student council, are girls. Five students from the council are to be randomly selected to participate in a student exchange to uddy-uddy-ville. 272. hat is the probability that at least 2 girls are selected? 273. hat is the probability that at least girl is selected? 27. hat is the probability that at most girls are selected? 0.986.000 0.9999355 0.87 36
Of the 5 students on this year s student council, 8 are girls. 6 students from the council are to be randomly selected to participate in a student exchange to uddy--sae. 275. hat is the probability that at least 2 girls are selected? 276. hat is the probability that at least girl is selected? 277. hat is the probability that at most 5 girls are selected? 0.965 0.999 0.99 Color Suit Non-Face Cards Face Cards Red earts A 2 3 5 6 7 8 9 0 J Q K Red Diamonds A 2 3 5 6 7 8 9 0 J Q K lack Clubs A 2 3 5 6 7 8 9 0 J Q K lack Spades A 2 3 5 6 7 8 9 0 J Q K Use the following 5 card hand descriptions to help you answer the questions below. Royal Flush Straight Flush Four of a kind A,K,Q,J,0 same suit Run of 5 same suit cards same rank Full ouse Flush Straight a pair and 3 of a kind All 5 same suit A run of 5, any suit 3 of a kind 3 cards same rank 2 pair wo pairs pair 2 cards same rank Calculate the following probabilities and round your answer to 6 decimals. 5 cards are dealt from a shuffled deck of 52 cards. hat is the probability of each event? 278. A full house made up of a pair of aces and 3 kings. 279. A flush made up of all hearts. 280. A royal flush made up of earts. (A,K,Q,J,0) 28. Four of a kind made up of aces. 282. Four of a kind made up of queens and an ace. 0.000009 0.00095 0.000000() 0.00008 0.000002 37
Calculate the following probabilities. A pizza store offers 5 different toppings on its pizzas. 283. Suppose a pizza has 5 toppings. ow many different pizzas can be made? 28. hat is the probability that the 5 randomly selected toppings will include ham, pineapple and feta cheese? 285. hat is the probability that the 5 randomly selected toppings will include pepperoni and bacon? Solution: 3 specific toppings choose 3 3 C 3 2 other toppings choose 2 2 C 2 3C 3 2 C 2 P (, PA, F ) 5C 5 3003 0.022 0.095 A pizza store offers 2 different toppings on its pizzas. 286. Suppose a pizza has toppings. ow many different pizzas can be made? 287. hat is the probability that the randomly selected toppings will include turkey, beets and spinach? 288. hat is the probability that the randomly selected toppings will include ground beef and sausage? 95 0.08 0.09 Calculate the following probabilities. Allie, etsy, Colleen, Debby and Edith are finalists in radio prize giveaway. 2 of them will be randomly selected to win prizes. hat is the probability of each event? 289. Allie is one of the winners. 290. Allie is a winner but etsy is not a winner. 29. Debby and Edith do not win Detailed solutions are available in the answer key. 0. 0.3 0.3 38
Enriched: Determine the probabilities and round your answer to 3 decimals. 292. P( of a kind in a 5 card hand) 293. P(Full house) P(3 of a kind and 2 of a kind) Solution Picking the equal cards Pick Rank 3 options pick 3C Pick Suit options pick C Picking other card must be different Pick Rank 2 options pick 2C Pick suit options pick C P( of a kind) 3C C 2 C C 52 C 5 0.0002 29. P(2 pair) 295. P(3 of a kind) 296. P( pair) 0.00 0.075 0.02 0.226 Calculate the following probabilities. Mr. Spray has 5 different Probabilities Retests to give randomly to his students. wo boys and 3 girls want to retest. Determine the probability of each event. 297. he 3 girls take the same test 298. Exactly 3 people take the same test. 299. All 5 people take a different test. 300. All 5 of them take the same test. 0.0 0.256 0.038 30. All the boys write the same test. All the girls write the same test but it is different than the boys. 302. Exactly 2 people take the same test and the 3 remaining take a different test. 0.002 0.006 0.38 39
inomial heorem and Probabilities he inomial heorem n C r (a)r (a) n r n is the # of options, r is the # chosen, a is the probability that an even occurs. Calculate the following probabilities. immy flips a biased coin and flips it times. he probability of heads is 0.6. Determine the following probabilities P(), P(3), P(2), P(), P(0). # OFEADS 3 3 2 3 2 2 3 2 2 2 0 # OF EADS P( EADS) C (0.6) 0.296 P(3 EADS) C 3 (0.6) 3 (0.) 0.356 P(2 EADS) C 2 (0.6) 2 (0.) 2 0.356 P( EAD) C (0.6) (0.) 3 0.536 P(0 EADS) C 0 (0.) 0.0256 ow many pathways to s ow many pathways to 3 s ow many pathways to 2 s ow many pathways to ow many pathways to 0 s C C 3 C 2 6 C C 0 (0.6)(0.6)(0.6)(0.6)0.296 (0.6)(0.6)(0.6)(0.)0.086 (0.6)(0.6)(0.)(0.6) 0.086 (0.6)(0.)(0.6) (0.6)0.086 (0.)(0.6)(0.6) (0.6) 0.086 (0.6)(0.6)(0.)(0.)0.0576 (0.6)( 0.)(0.6)( 0.)0.0576 (0.6)( 0.)( 0.)(0.6)0.0576 (0.)(0.6)(0.6)( 0.)0.0576 (0.)(0.6)( 0.)(0.6)0.0576 (0.)( 0.)(0.6)(0.6)0.0576 (0.6)(0.)(0.)(0.)0.038 (0.)(0.6)(0.)(0.)0.038 (0.)(0.)(0.6)(0.)0.038 (0.)(0.)(0.)(0.6)0.038 (0.)(0.)(0.)(0.)0.0256 NOICE: (0.6) 0.296 NOICE: (0.6) 3 (0.) 0.086 NOICE: (0.6) 2 (0.) 2 0.0576 NOICE: (0.6) (0.) 3 0.038 NOICE: (0.) 0.0256 Remember: Order of fractions does not matter. 0
Calculate the following probabilities. Round your answer to 3 decimals. A fair coin is tossed 5 times. Determine the probability of each event. 303. P(0 heads) 30. P( heads) 305. P(2 heads) 306. P(3 heads) 307. P( head) 308. P(5 heads) 0.03 0.56 0.33 0.33 0.56 0.03 309. P(at least head) 30. P(at most heads) 0.969 0.969 ow can I get my calculator to do the same thing A coin is tossed 5 times. Determine the probability of each event. *Pdfbin(N,P,S) NNumber of events PProbability of success SNumber of successful events Calculates probability of a single event. *Cdfbin(N,P,S) NNumber of events PProbability of success She sum of the first S successful events Calculates sum of probabilities. 3. P( heads) Pdfbin(5,0.5,) 32. P(3 eads) 33. P(0 eads) 3. P(At most 3 heads) Cdfbin(5,0.5,3) 35. P(At most heads) 36. P(At least least heads) *Press SA Choose F DISRIUION Choose (0)pdfbin or ()cdfbin. IMPORAN hese options can only be used to calculate probabilities for a binomial distribution. 0.56 0.33 0.03 0.83 0.969 0.88 A biased coin is tossed 5 times, where P().6. Determine the probability of each event. 37. 5 tails 38. tails 39. 3 tails 320. 3 heads 32. heads 322. 5 heads 0.078 0.259 0.36 0.230 0.077 0.00 323. If you have at least tails, what is the probability that you have 5? 32. omit 0.23
Challenge #22: he probability of the Vancouver Canucks winning any game against any opponent is 0.60 Determine the probability that the Canucks win 6 of the next seven games (Any order). Challenge #23: he probability of a Canucks win in any game of a best of seven game series is 60%. Determine the probability that the Canucks win a best of 7 series in exactly 6 games. Calculate the following probabilities. Round your answer to 3 decimals. he probability of the Vancouver Canucks winning any game against any opponent is 60%. 325. P(Canucks win 6 of 7 games) 326. P(Canucks win 5 of 7games) 327. P(Canucks win 7 of 7games) 0.3 0.26 0.0279 he probability of a Canucks win in any game of a best of seven game series is 60%. 328. P(Canucks win in the 6 th 329. P(Canucks win the best of 7 330. P(Canucks win the best of 7 game) series in the 5 th game) series in the 7 th game) Solution: his is a tricky one o win in exactly 6 games means that the 6 th game must be a win. his means that the st 5 games resulted in 3 & 2L P(3,2L) and P(in 6 th ) 5 C 3 (0.6) 3 (0.) 2 (0.6) 0.207 0.207 0.66 2
he probability of a Canucks win in any game of a best of seven game series is 60%. 33. P(Canucks lose in games) 332. P(Canucks lose in 7 games) 333. P(Canucks lose in 6 games) 0.026 0. 0.092 3
Questions 30-360 are good review for the test! 33. A biased coin with P(eads)0.7 was tossed 5 times. Find P(3) 335. 2 regular die are rolled. Find P(sum is at most 0) 336. A biased coin with P(eads)0.7 was tossed 3 times. Find P(2 & ) 337. he probability that your new I- ook last 3 years is 0.9 and that it will last 5 years is 0.3. After 3 years what is the chance that your I-ook will break down in the next two years? 338. In a recent survey of grade 2 students, it was found that 70% took math and 50% took chemistry. If 80% took math or chemistry, what percent of students took math only? 339. immy is planning to role a die 6 times. After he rolled 2 fives and 2 threes, he wonders what the probability will be that the 6 roles will result in 2 fives and threes. 30. A regular die is rolled twice. Determine the probability that the first toss is greater than and the second is less than. 3. 80% of girls want to go into business and 30% want to go into education. 20% want neither. Find P(Girl pursue at most one career) 32. A certain experiment has only outcomes: A,, C, D. he probability of each outcome is twice the probability of the following outcome. Find P() 33. he probability of the Vancouver Canucks winning any game against any opponent is 60%. Find P(Canucks win 5 of 7games) 3. A pizza store offers 2 different toppings on its pizzas. hat is the probability that the randomly selected toppings will include ground beef and sausage? 35. (optional) A six-sided die is rolled 00 times. rite a formula using cdfbin or pdfbin for P(at most 99 sixes)
36. here are 3 black marbles, white marbles and 3 striped marble in the bag. e takes out one marble, looks at it, puts it back and then randomly draws another marble. Find P( &S) 37. Of the 5 students on this year s student council, 8 are girls. 6 students from the council are to be randomly selected to participate in a student exchange to uddy--sae. Find P(at least girl is selected). 38. A computer supply store buys 0% of computer chips from X-Chips and 60% from Y-Chips. On average 6% of the X-Chips are faulty and 5% of the Y-Chips are faulty. If a randomly selected chip is faulty, what is the probability that X-Chips made it? 39. A single card is selected from a deck of 52 cards. Determine P(Club or jack) 350. A biased -sided die is weighted so that it returns a, 0% of the time. Determine P(2) 35. Find P(2 fours, 2 fives, any other card) when 5 cards are drawn from a regular deck. 352. wo cards are drawn without replacement from a shuffled deck of 52 cards. Find P( st Club, 2 nd 8 of Clubs). 353. he probability of the Vancouver Canucks winning any game against any opponent is 60%. Find P(Canucks win a best of 7 game series in 7 games) 35. wo cards are drawn without replacement from a shuffled deck of 52 cards. Find P(King of Spades is the 2 nd card). Answers can be found in the answer key. 5
Answers to Probabilities Notes ) 0.5 2) Answers will vary with each experiment, but the probability will be approximately 0.5. 3) 6) 7) 0 3 ) 5) 8) 9) 2 26 52 0) 3 2 0 ) 2) 3) ) 5) 6) 26 3 3 3 3 3 7) h,h2,h3,h,h5,h6 7 3 8) 9) 20) 2) t,t2,t3,t,t5,t6 2 2 22),2,3,,2,22,23,2 3 9 27) hhh,hht,hth,thh 23) 2) 25) 26) 3,32,33,3,,2,3, 8 6 2 ttt,tth,tht,htt 28) 7 7 3) 0.7 32) 0.3 33) 3) 5 29) 30) 8 8 8 35) 0.6 36) 0. 37) 8 38) 2 39) 0.2 0) 0.6 ) 8 2) 96 3) deck of 3 ) 5) 6) cards 2 3 7) 50 cards since 2 cards 6 2 50) 0 8) 9) 5) have already been used 25 25 2 52) 35 56) 0.25 53) 5) 55) 6 36 2 6 57) 0.0325 58) 0.0039 59) 0.000 60) 0.0000 7 5 6) 62) 0 0 63) 3 5 3 9 7 6 6) 65) 66) 67) 68) 69) 70) 0 0 0 0 0 0 0 0 7) 3 3 9 6 72) 73) 7) 75) 76) 77) 78) 5 7 3 3 3 3 3 79) 8 7 2 9 85) 86) 0 80) 8) 82) 83) 8) 6 3 3 3 3 3 87) 0 5 89) 0.8 90) 0.5 9) 0.9 92) 0.8 93) 0.78 88) 3 3 9) 0.6 95) 0.3 96) 0. 97) 0.9 98) 0.3 99) 0.7 00) 0) 2 3 02) 5 0) 05) 7 06) 07) 08) 0.5 09) 0. 3 03) 26 3 26 2 52 0) 0.25 ) 0.95 2) 0. 3) 0.3 ) 0.9 5) 0.5 6) 0.3 7) 30% 8) 0% 9) 0% 20) 20% 2) 5% 22) 35% 23) 20% 2) 30% 25) 8 26) 3 8 27) 3 8 28) 7 8 29) 0.333 30) 0.533 3) 0.867 77) 5 5 0 + 0 5 5 78) 5 5 + n n + n + n 5 + n 5 + n 79) x x + 0 0 x + 9 + 0 x + 0 9 x + 9 6
Alli, etsy, Colleen, Debby and Edith are finalists in radio prize giveaway. 2 of them will be randomly selected to win prizes. hat is the probability of each event? 290) Alli is one of the winners 29) Alli is a winner but etsy is not 292) Debby and Edith do not win A one way to pick A other options pick A one way to pick A do not pick it 3 other options pick E do not pick it D do not pick it 3 other options pick 2 P ( A) C 5 C 2 C 0 0. P ( Anot ) C C 5 C 3 2 3 0 0.3 3C P ( Anot ) C 2 5 2 3 0 0.3 NOE: his Question can be done by listing the sample set and counting. Much easier!! A,AC,AD,AE C,D,E CD,CE DE A,AC,AD,AE C,D,E CD,CE DE A,AC,AD,AE C,D,E CD,CE DE A,AC,AD,AE C,D,E CD,CE DE Sample set out of 0 3 out of 0 3 out of 0 293) Full ouse: Picking 3 equal cards Pick Rank 3 options pick 3C 3 Pick Suit options pick 3 C 3 Picking 2 equal cards Different than the 3 above Pick Rank 2 options pick 2C 2 Pick suit options pick 2 C 26 29) 2 Pairs Picking the 2 pairs Pick Rank 3 options pick2 3C 278 Pick Suit of st pair options pick 2 C 26 Pick Suit of 2 nd pair options pick 2 C 26 Picking the 5 th card must be different than pairs Pick Rank options pick C Pick Suit options pick C 3 C C 3 2 C C 2 52 C 5 0.00 3 C 2 C 2 C 2 C C 52 C 5 0.75 295) 3 of a kind in a 5 card hand Picking the 3 or a kind Pick Rank 3 options pick 3C 3 Pick Suit of pair options pick 3 C 3 Pick 2 other cards must be different Pick rank 2 options pick 2 2C 266 Pick suit options pick C Pick suit options pick C 296) pair in a 5 card hand Picking the pair Pick Rank 3 options pick 3C 3 Pick Suit of pair options pick 2 C 26 Pick 3 different cards must be different Pick rank 2 options pick 3 2C 3220 Pick suit of st card options pick C Pick suit of 2 nd card options pick C Pick suit of 3 rd card options pick C 3 C C 3 2 C 2 C C 52 C 5 0.02 C C C C C C 3 2 2 3 0.226 52 C 5 7