Outcomes: The outcomes of this experiment are yellow, blue, red and green.


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1 (Adapted from 1. Sample Space The sample space of an experiment is the set of all possible outcomes of that experiment. The sum of the probabilities of the distinct outcomes within a sample space is 1. Ex 1: What is the probability of each outcome when a coin is tossed? Outcomes: The outcomes of this experiment are head and tail. P(Head) = 0.5 P(Tail) = 0.5 The events Head and Tail are mutually exclusive and exhaustive and thus they are complementary events. P(Head) + P(Tail) = 1 The sample space of Experiment 1 is: {head, tail} Ex 2: A spinner has 4 equal sectors coloured yellow, blue, green and red. What is the probability of landing on each colour after spinning this spinner? Outcomes: The outcomes of this experiment are yellow, blue, red and green. P(yellow) = 0.25 P(blue) = 0.25 P(red) = 0.25 P(green) = 0.25 The events yellow, blue, red and green are mutually exclusive and exhaustive and thus they are complementary events. P(Y) + P(B) + P(R) + P(G) = 1 The sample space of Experiment 2 is: {Yellow, Blue, Red, Green}
2 Exercises 1. What is the sample space for choosing an odd number from 1 to 11 at random? A: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 B: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} C: {1, 3, 5, 7, 9 11} 2. What is the sample space for choosing a prime number less than 15 at random? A: {2, 3, 5, 7, 11, 13, 15} B; {2, 3, 5, 7, 11, 13} C: {2, 3, 5, 7, 9, 11, 13} D: All of the above. 3. What is the sample space for choosing 1 jellybean at random from a jar containing 5 red, 7 blue and 2 green jellybeans? A: {5, 7, 2} B: {5 red, 7 blue, 2 green} C: {red, blue, green} 4. What is the sample space for choosing 1 letter at random from 5 vowels? A: {a, e, i, o, u} B: {v, o, w, e, l} C: {1, 2, 3, 4, 5} 5. What is the sample space for choosing 1 letter at random from the word DIVIDE? A: {d, i, v, i, d, e} B: {1, 2, 3, 4, 5, 6} C: {d, i, v, e}
3 2. Addition Rule for calculating probabilities To find the probability of event A or B, we must first determine whether the events are mutually exclusive or nonmutually exclusive. Then we can apply the appropriate Addition Rule: Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B) Addition Rule 2: When two events, A and B, are nonmutually exclusive, there is some overlap between these events. The probability that A or B will occur is the sum of the probability of each event, minus the probability of the overlap. P(A or B) = P(A) + P(B)  P(A and B) Examples Ex 1: A single 6sided die is rolled. What is the probability of rolling a 2 or a 5? Probabilities: These events are mutually exclusive since they cannot occur at the same time. P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 2/6 = 1/3 Ex 2: In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls achieved an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student? Probabilities: These events are NOT mutually exclusive since it is possible for a student, chosen randomly, to be a girl and an A student. P(girl or A) = P(girl) + P(A)  P(girl and A) = 13/30 + 9/305/30 = 17/30
4 Exercises 1. A day of the week is chosen at random. What is the probability of choosing a Monday or Tuesday? A: 1/7 B: 1/14 C: 2/7 2. In a pet store, there are 6 puppies, 9 kittens, 4 gerbils and 7 parakeets. If a pet is chosen at random, what is the probability of choosing a puppy or a parakeet? A: 15/26 B: 1/2 C: 11/26 3. The probability of a New York teenager owning a skateboard is 0.37, of owning a bicycle is 0.81 and of owning both is If a New York teenager is chosen at random, what is the probability that the teenager owns a skateboard or a bicycle? A: 1.18 B: 0.7 C: A number from 1 to 10 is chosen at random. What is the probability of choosing a 5 or an even number? A: 3/5 B: 1/2 C: 1/5 D: All of the above. 5. A single 6sided die is rolled. What is the probability of rolling a number greater than 3 or an even number? A: 1 B: 2/3 C: 5/6
5 3. Independent events Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Some examples of independent events are: 1. Landing on heads after tossing a coin AND rolling a 5 on a single 6sided die. 2. Choosing a marble from a jar AND landing on heads after tossing a coin. 3. Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card. 4. Rolling a 4 on a single 6sided die, AND then rolling a 1 on a second roll of the die. To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by AND. Multiplication Rule 1: When two events, A and B, are independent, the probability of both occurring is: P(A and B) = P(A) x P(B) Examples Ex 1: A coin is tossed and a single 6sided die is rolled. Find the probability of tossing a head and rolling a 3 on the die. NB: These are independent events. Probabilities: P(head) = 1/2 ; P(3) = 1/6 P(head and 3) = P(head) x P(3) = 1/2 x 1/6 = 1/12 Multiplication Rule 1 can be extended to work for three or more independent events that occur in sequence as follows: Ex 2: A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza? Probabilities: P(student 1 likes pizza) = 9/10 P(student 2 likes pizza) = 9/10
6 P(student 3 likes pizza) = 9/10 P(student 1 and student 2 and student 3 like pizza) = 9/10 x 9/10 x 9/10 Exercises = 729/1000 = = 72.9% 1. Spin a spinner numbered 1 to 7, and toss a coin. What is the probability of getting an odd number on the spinner and a tail on the coin? A: 3/14 B: 2/7 C: 5/14 2. A jar contains 6 red balls, 3 green balls, 5 white balls and 7 yellow balls. Two balls are chosen from the jar, with replacement. What is the probability that both balls chosen are green? A: 6/441 B: 1/49 C: 2/49 D: None of the above 3. In Exercise 2 above, what is the probability of choosing a red and then a yellow ball? A: 2/21 B: 3/21 C: 13/63 D: All of the above. 4. Four cards are chosen from a standard deck of 52 playing cards with replacement. What is the probability of choosing 4 hearts in a row? A: 13/52 B: 1/16 C: 1/256
7 5. A nationwide survey showed that 65% of all children in the United States dislike eating vegetables. If 4 children are chosen at random, what is the probability that all 4 dislike eating vegetables? (Round your answer to the nearest percent.) A: 18% B: 260% C: 2% 4. Dependent events Definition: Two events are dependent if the outcome or occurrence of the first event A affects the outcome or occurrence of the second event B, so that the probability of event B is changed. This leads to the following concept: Definition: The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B A) [pronounced as The probability of event B given A ]. Multiplication Rule 2: When two events, A and B, are dependent, the probability of both occurring is: P(A and B) = P(A) x P(B A) Examples Ex 1: A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? Probabilities: P(queen on first pick) = 4/52 P(jack on 2nd pick queen) = 4/51 P(queen and jack) = 4/52 x 4/51 = 16/2652 = 4/663 = 0.6%
8 Ex 2: Mr Moser, who has changed schools to teach at a coed school, needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly chooses one student. He then chooses a second student from those still seated. What is the probability that both students chosen are girls? Probabilities: P(Girl 1 and Girl 2) = P(Girl 1) x P(Girl 2 Girl 1) = 18/30 x 17/29 = 306/870 = 51/145 = 35.2% Ex 3: Three cards are chosen at random from a deck of 52 cards without replacement. What is the probability of choosing 3 aces? Probabilities: P(3 aces) = 4/52 x 3/51 x 2/50 = 24/ = 1/5525 = 0.02% Exercises 1. Two cards are chosen at random from a deck of 52 cards without replacement. What is the probability of choosing two kings? A: 4/663 B: 1/221 C: 1/69 2. Two cards are chosen at random from a deck of 52 cards without replacement. What is the probability that the first card is a jack and the second card is a ten? A: 3/676 B: 1/169 C: 4/663
9 3. On a math test, 5 out of 20 students got an A. If three students are chosen at random without replacement, what is the probability that all three got an A on the test? A: 1/114 B: 25/1368 C: 3/ Three cards are chosen at random from a deck of 52 cards without replacement. What is the probability of choosing an ace, a king, and a queen in order? A: 1/2197 B: 8/5525 C: 8/ A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that all four walk to school? A: 343/93960 B: 1/783 C: 7/6750
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