MATH 2000 TEST PRACTICE 2 1. Maggie watched 100 cars drive by her window and compiled the following data: Model Number Ford 23 Toyota 25 GM 18 Chrysler 17 Honda 17 What is the empirical probability that the next car is (a) A Ford? (b) Not a Toyota? (c) A Chrysler or a GM? (d) A Rolls Royce? 2. In a class of 100 students, 75 take statistics, 21 take calculus and 13 take both subjects. What is the probability that a randomly selected student takes neither statistics nor calculus? 3. In an English class there 12 juniors and 15 seniors; 7 of the juniors are males and 5 of the seniors are females. If a student is selected at random, find the probability of selecting the following: (a) A senior or a female (b) A junior or a senior 4. In an election 47% of eligible voters did not vote. If three eligible voters are selected at random find: (a) The probability that none of them voted in the election. (b) The probability that at least one of the three voted in the election. 5. If two cards are selected from a 52 card deck without replacement, find these probabilities: (a) Both are kings (b) The cards are different suits 6. In a board of directors composed of eight people, in how many ways can one chief executive officer, one director, and one treasurer be selected?
7. At a small college, the probability that a student takes physics and sociology is 0.18. The probability that a student takes sociology is 0.72. Find the probability that a student is taking physics, given that he is taking sociology. To get credit you must show the formula you use to get your answer. 8. In how many ways can a committee of four people be selected from a pool of ten people? 9. In how many ways can a jury of six women and six men be selected from nine women and twelve men? 10. A bag contains five red balls and seven white balls. If you select four balls at random without replacement, find the probability that you get two red balls and two white balls. 11. A box contains four $1 bills, five $5 bills, one $10 bill and six $100 bills. Construct a probability distribution for the experiment of selecting one bill at random from the box. 12. Find the mean, variance and standard deviation for the distribution shown. X 1 2 3 4 5 PX ( ) 0.3 0.15 0.36 0.12 0.07 13. Dogdale College needs to raise money to buy a computer. They decide to conduct a raffle. A cash prize of $4000 is to be awarded. If they sell 3000 tickets at $3 each find the expected gain if you buy one ticket. (There will be only one winning ticket.) You must set up a probability distribution to get credit. 14. It is reported that 77% of workers aged 16 and over drive to work alone. Choose 8 workers at random. Find the probability that: a) all drive to work alone. b) more than one half drive to work alone c) exactly three drive to work alone
Name: MATH 2000 PRACTICE TEST 2 ANSWERS 1. Maggie watched 100 cars drive by her window and compiled the following data: Model Number Ford 23 Toyota 25 GM 18 Chrysler 17 Honda 17 What is the empirical probability that the next car is (a) A Ford? Empirical probability is what you would expect the probability to be based on past events. 23 100 (b) Not a Toyota? 75 cars were not Toyotas, so P(not a Toyota) = 75 3 100 4 25 75 3 You could also say P(not a Toyota) = 1 P(is a Toyota) = 1 100 100 4 (c) A Chrysler or a GM? 18 17 35 7 100 100 20 (d) A Rolls Royce? Based on past observations it is not possible for the next car to be a Rolls Royce. 0 2. In a class of 100 students, 75 take statistics, 21 take calculus and 13 take both subjects. What is the probability that a randomly selected student takes neither statistics nor calculus? Taking neither statistics nor calculus is the complement of taking one or the other. Use P( S or C) P( S) P( C) P( S and C)
P( S or C) P( S) P( C) P( S and C) 75 21 13 100 100 100 83.83 100 P(neither) = 1.83 =.17 3. In an English class there 12 juniors and 15 seniors; 7 of the juniors are males and 5 of the seniors are females. If a student is selected at random, fined the probability of selecting the following: (a) A senior or a female It can be helpful to put the information into a chart. Male Female Junior 7 5 12 Senior 10 5 15 17 10 27 P(senior or female) = P(senior) + P(female) P(senior and female) 15 10 5 20 27 27 27 27 (b) A junior or a senior The events being a junior and being a senior are mutually exclusive so, 12 15 P(junior or senior) = P(junior) + P(senior) 1 27 27 Note that you may say that the answer is one because it is certain that a student will be either a junior or a senior. 4. In an election 47% of eligible voters did not vote. If three eligible voters are selected at random find: (a) The probability that none of them voted in the election. In this type of problem you assume that the events are independent, in which case you multiply probabilities. P(none voted) = (.47)(.47)(.47) =.103823 Can also use the binompdf(3,.47, 3) =.103823 (b) The probability that at least one of the three voted in the election. At least one is the complement of none P(at least one voted) = 1 P(none voted) =1.103823 =.896177
5. If two cards are selected from a 52 card deck without replacement, find these probabilities: (a) Both are kings P(both are kings) = P(first is a king)p(second is a king, given that the first is a king) 4 3 1 52 51 221 (b) The cards are different suits The first card can be any card. Whatever suit it is there will be 51 cards left of which 39 will be in the other three suits. P(different suits) = 52 39 13 52 51 17 6. In a board of directors composed of eight people, now many ways can one chief executive officer, one director, and one treasurer be selected? By the counting principle it is 876 336 You may also figure it as 8 P 3. You use P rather than C because picking the same three people in a different order constitutes a different selection. 7. At a small college, the probability that a student takes physics and sociology is 0.18. The probability that a student takes sociology is 0.72. Find the probability that a student is taking physics, given that he is taking sociology. To get credit you must show the formula you use to get your answer. P(A and B) You use the formula P( A/ B) P(B) P(physics and sociology) P(physics sociology) P(sociology).18 1.25.72 4 8. In how many ways can a committee of four people be selected from a pool of ten people? 10C4 210 You use C rather than P because picking the same four people in a different order constitutes the same committee.
9. In how many ways can a jury of six women and six men be selected from nine women and twelve men? The six women can be chosen in 9 C 6 ways. The six men can be chosen in 12 C 6 ways. By the counting principle the number of ways six women and six men can be chosen C C 84924 77616. is 9 6 12 6 10. A bag contains five red balls and seven white balls. If you select four balls at random without replacement, find the probability that you get two red balls and two white balls. The two red balls can be chosen in 5 C 2 ways. The two white balls can be chosen in 7C 2 ways. By the counting principle the two red balls and two white balls can be chosen in 5 C 2 * 7 C 2 ways. There are 12 balls so the 4 balls can be chosen in 12 C 4 ways. C C 10 21 P(2 red balls and 2 white balls) = 5 2 7 2 0.424 C 495 12 4 11. A box contains four $1 bills, five $5 bills, one $10 bill and six $100 bills. Construct a probability distribution for the experiment of selecting one bill at random from the box. There are 16 bills so, 4 1 P($1) ; 16 4 5 P($5) ; 16 1 P($10) ; 16 6 3 P($100) 16 8 X PX ( ) 1 5 10 100 1 5 1 3 4 16 16 8 12. Find the mean, variance and standard deviation for the distribution shown. X 1 2 3 4 5 PX ( ) 0.3 0.15 0.36 0.12 0.07 XP( X ) 1(.3) 2(.15) 3(.36) 4(.12) 5(.07) 2.51 2 2 2 2 2 2 2 2 2 X P( X ) 1 (.3) 2 (.15) 3 (.36) 4 (.12) 5 (.07) 2.51 1.5099 2 1.2288
13. Dogdale College needs to raise money to buy a computer. They decide to conduct a raffle. A cash prize of $4,000 is to be awarded. If they sell 3,000 tickets at $3 each find the expected gain if you buy one ticket. (There will be only one winning ticket.) You must set up a probability distribution to get credit. If you win your gain is $3,997. If you do not win your gain is $3. The probability 1 2999 that you win is and the probability that you do not win is 3000 3000. The probability distribution is X 3 3,997 PX ( ) 2,999 1 3,000 3,000 2,999 1 Expected gain = XP( X ) 3( ) 3997( ) $1.67 3000 3000 14. It is reported that 77% of workers aged 16 and over drive to work alone. Choose 8 workers at random. Find the probability that: 8! 8 0 ( ) (8).77.23 0.124 0!8! a) all drive to work alone. Exactly 8 uses pdf P x P Or using the calculator n = 8, p =.77, X = 8; Push the following buttons: 2 nd vars; down to binompdf(n, p, x) = binompdf(8,.77, 8) = 0.124 b) more than one half drive to work alone. P( x) P(5, 6, 7, or 8) 0.912 = 8! 5 3 8! Px ( ).77.23 6 2 8!.77.23 7 1 8!.77.23.77 8.23 0 3!5! 2!6! 1!7! 0!8! Or using the calculator n = 8, p =.77, X = 5, 6, 7, 8; Binomials only measure from left to right. So we will use 1 P(0, 1, 2, 3, 4). To do this push the following buttons: 1 2 nd vars; down to binomcdf(n, p, x) = 1 binomcdf(8,.77, 4) = 0.912 Notice that the binomcdf was used to get the cumulative value. 8! 3 5 c) exactly three drive to work alone. P( x) P(3).77.23 0.017 5!3! Or using the calculator n = 8, p =.77, X = 3; Push the following buttons: 2 nd vars; down to binompdf(n, p, x) = binompdf(8,.77, 3) = 0.017