Consider the birthdays of the students in a class of size r. Assume that the year

Bowl A contains three red and two white chips, and bowl B contains four red and three white chips. A chip is drawn at random from bowl A and transferred to bowl B. Compute the probability of then drawing a red chip from bowl B.

A common test for AIDS is called the ELISA (enzyme-linked immunosorbent assay) test. Among 1 million people who are given the ELISA test, we can expect results similar to those given in the following table: If one of these 1 million people is selected randomly, find the following probabilities: (a) \(P\left(B_{1}\right)\), (b) \(P\left(A_{1}\right)\), (c) \(P\left(A_{1} \mid B_{2}\right)\), (d) \(P\left(B_{1} \mid A_{1}\right)\). (e) In words, what do parts (c) and (d) say? Equation Transcription: Text Transcription: P(B_1) P(A_1) P(A_1?B_2) P(B_1?A_1)

The following table classifies 1456 people by their gender and by whether or not they favor a gun law. Compute the following probabilities if one of these 1456 persons is selected randomly: (a) \(P\left(A_{1}\right)\), (b) \(P\left(A_{1} \mid S_{1}\right)\), (c) \(P\left(A_{1} \mid S_{2}\right)\). (d) Interpret your answers to parts (b) and (c). Equation Transcription: Text Transcription: P(A_1) P(A_1?S_1) P(A_1?S_2)

PROBLEM 4E Two cards are drawn successively and without replacement from an ordinary deck of playing cards. Compute the probability of drawing (a) Two hearts. (b) A heart on the first draw and a club on the second draw. (c) A heart on the first draw and an ace on the second draw. Hint: In part (c), note that a heart can be drawn by getting the ace of hearts or one of the other 12 hearts.

Let \(A_{1}\) and \(A_{2}\) be the events that a person is left eye dominant or right-eye dominant, respectively. When a person folds his or her hands, let \(B_{1}\) and \(B_{2}\) be the events that the left thumb and right thumb, respectively, are on top. A survey in one statistics class yielded the following table: If a student is selected randomly, find the following probabilities: (a) \(P\left(A_{1} \cap B_{1}\right)\), (b) \(P\left(A_{1} \cup B_{1}\right)\), (c) \(P\left(A_{1} \mid B_{1}\right)\), (d) \(P\left(B_{2} \mid A_{2}\right)\). (e) If the students had their hands folded and you hoped to select a right-eye-dominant student, would you select a "right thumb on top" or a "left thumb on top" student? Why? Equation Transcription: Text Transcription: A_1 A_2 B_1 B_2 P(A_1 cap B_1) P(A_1 cup B_1) P(A_1?B_1) P(B_2?A_2)

PROBLEM 6E A researcher finds that, of 982 men who died in 2002, 221 died from some heart disease. Also, of the 982 men, 334 had at least one parent who had some heart disease. Of the latter 334 men, 111 died from some heart disease. A man is selected from the group of 982. Given that neither of his parents had some heart disease, find the conditional probability that this man died of some heart disease.

PROBLEM 7E An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?

PROBLEM 8E An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting a single ball at random from the urn without replacement. The person who selects the third WIN ball wins the game. It does not matter who selected the first two WIN balls. (a) If you draw first, find the probability that you win the game on your second draw. (b) If you draw first, find the probability that your opponent wins the game on his second draw. (c) If you draw first, what is the probability that you win? Hint: You could win on your second, third, fourth, . . . , or tenth draw, but not on your first. (d) Would you prefer to draw first or second?Why?

PROBLEM 10E A single card is drawn at random from each of six well-shuffled decks of playing cards. Let A be the event that all six cards drawn are different. (a) Find P(A). (b) Find the probability that at least two of the drawn cards match.

You are a member of a class of 18 students. A bowl contains 18 chips: 1 blue and 17 red. Each student is to take 1 chip from the bowl without replacement. The student who draws the blue chip is guaranteed an A for the course. (a) If you have a choice of drawing first, fifth, or last, which position would you choose? Justify your choice on the basis of probability. (b) Suppose the bowl contains 2 blue and 16 red chips. What position would you now choose?

PROBLEM 11E Consider the birthdays of the students in a class of size r. Assume that the year consists of 365 days. (a) How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacement)? (b) The same as part (a), except requiring that all the students have different birthdays (without replacement)? (c) If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that at least two students have the same birthday? (d) For what value of r is the probability in part (c) about equal to 1/2? Is this number surprisingly small? Hint: Use a calculator or computer to find r.

Paper is often tested for “burst strength” and “tear strength.” Say we classify these strengths as low, middle, and high. Then, after examining 100 pieces of paper, we find the following: If we select one of the pieces at random, what are the probabilities that it has the following characteristics: (a) \(A_{1}\) ? (b) \(A_{3} \cap B_{2}\) ? (c) \(A_{2} \cup B_{3}\) ? Equation Transcription: Text Transcription: A_1 A_3 cap B_2 A_2 cup B_3

PROBLEM 15E An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is 151/300. How many red balls are in the second urn?

PROBLEM 13E In the gambling game “craps,” a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2, 3, or 12. If the sum is 4, 5, 6, 8, 9, or l0, that number is called the bettor’s “point.” Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7, the bettor wins. (a) List the 36 outcomes in the sample space for the roll of a pair of dice. Assume that each of them has a probability of 1/36. (b) Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11, P(7 or 11). (c) Given that 8 is the outcome on the first roll, find the probability that the bettor now rolls the point 8 before rolling a 7 and thus wins. Note that at this stage in the game the only outcomes of interest are 7 and 8. Thus find P(8 |7 or 8). (d) The probability that a bettor rolls an 8 on the first roll and then wins is given by P(8)P(8 | 7 or 8). Show that this probability is (5/36)(5/11). (e) Show that the total probability that a bettor wins in the game of craps is 0.49293. Hint: Note that the bettor can win in one of several mutually exclusive ways: by rolling a 7 or an 11 on the first roll or by establishing one of the points 4, 5, 6, 8, 9, or 10 on the first roll and then obtaining that point on successive rolls before a 7 comes up.

A common test for AIDS is called the ELISA (enzyme-linked immunosorbent assay) test. Among 1 million people who are given the ELISA test, we can expect results similar to those given in the following table: B1: Carry B2: Do Not AIDS Virus Carry Aids Virus Totals A1: Test Positive 4,885 73,630 78,515 A2: Test Negative 115 921,370 921,485 Totals 5,000 995,000 1,000,000 If one of these 1 million people is selected randomly, find the following probabilities: (a) P(B1), (b) P(A1), (c) P(A1 | B2), (d) P(B1 | A1). (e) In words, what do parts (c) and (d) say?

The following table classifies 1456 people by their gender and by whether or not they favor a gun law. Male (S1) Female (S2) Totals Favor (A1) 392 649 1041 Oppose (A2) 241 174 415 Totals 633 823 1456 Compute the following probabilities if one of these 1456 persons is selected randomly:(a) P(A1),(b) P(A1 | S1),(c) P(A1 | S2). (d) Interpret your answers to parts (b) and (c).

Let A1 and A2 be the events that a person is lefteye dominant or right-eye dominant, respectively. When a person folds his or her hands, let B1 and B2 be the events that the left thumb and right thumb, respectively, are on top. A survey in one statistics class yielded the following table: B1 B2 Totals A1 5 7 12 A2 14 9 23 Totals 19 16 35 If a student is selected randomly, find the following probabilities: (a) P(A1 B1), (b) P(A1 B1), (c) P(A1 | B1), (d) P(B2 | A2). (e) If the students had their hands folded and you hoped to select a right-eye-dominant student, would you select a right thumb on top or a left thumb on top student? Why?

Two cards are drawn successively and without replacement from an ordinary deck of playing cards. Compute the probability of drawing (a) Two hearts. (b) A heart on the first draw and a club on the second draw. (c) A heart on the first draw and an ace on the second draw. Hint: In part (c), note that a heart can be drawn by getting the ace of hearts or one of the other 12 hearts.

Suppose that the alleles for eye color for a certain male fruit fly are (R, W) and the alleles for eye color for the mating female fruit fly are (R, W), where R and W represent red and white, respectively. Their offspring receive one allele for eye color from each parent. (a) Define the sample space of the alleles for eye color for the offspring. (b) Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offsprings eyes look white, what is the conditional probability that it has two white alleles for eye color?

A researcher finds that, of 982 men who died in 2002, 221 died from some heart disease. Also, of the 982 men, 334 had at least one parent who had some heart disease. Of the latter 334 men, 111 died from some heart disease. A man is selected from the group of 982. Given that neither of his parents had some heart disease, find the conditional probability that this man died of some heart disease.

An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?

An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting a single ball at random from the urn without replacement.The person who selects the third WIN ball wins the game. It does not matter who selected the first two WIN balls. (a) If you draw first, find the probability that you win the game on your second draw. (b) If you draw first, find the probability that your opponent wins the game on his second draw. (c) If you draw first, what is the probability that you win? Hint: You could win on your second, third, fourth, . . . , or tenth draw, but not on your first. (d) Would you prefer to draw first or second? Why?

An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if the ball numbered m is the mth ball selected. Let the event Ai denote a match on the ith draw, i = 1, 2, 3, 4. (a) Show that P(Ai) = 3! 4! for each i. (b) Show that P(Ai Aj) = 2! 4! , i = j. (c) Show that P(Ai Aj Ak) = 1! 4! , i = j, i = k, j = k. (d) Show that the probability of at least one match is P(A1 A2 A3 A4) = 1 1 2! + 1 3! 1 4! . (e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is P(A1 A2 An) = 1 1 2! + 1 3! 1 4! ++ (1)n+1 n! = 1  1 1 1! + 1 2! 1 3! ++ (1)n n!  . (f) What is the limit of this probability as n increases without bound?

A single card is drawn at random from each of six well-shuffled decks of playing cards. Let A be the event that all six cards drawn are different. (a) Find P(A). (b) Find the probability that at least two of the drawn cards match.

Consider the birthdays of the students in a class of size r. Assume that the year consists of 365 days. (a) How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacement)? (b) The same as part (a), except requiring that all the students have different birthdays (without replacement)?(c) If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that at least two students have the same birthday? (d) For what value of r is the probability in part (c) about equal to 1/2? Is this number surprisingly small? Hint: Use a calculator or computer to find r.

You are a member of a class of 18 students. A bowl contains 18 chips: 1 blue and 17 red. Each student is to take 1 chip from the bowl without replacement. The student who draws the blue chip is guaranteed an A for the course. (a) If you have a choice of drawing first, fifth, or last, which position would you choose? Justify your choice on the basis of probability. (b) Suppose the bowl contains 2 blue and 16 red chips. What position would you now choose?

In the gambling game craps, a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2, 3, or 12. If the sum is 4, 5, 6, 8, 9, or l0, that number is called the bettors point. Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7, the bettor wins. (a) List the 36 outcomes in the sample space for the roll of a pair of dice. Assume that each of them has a probability of 1/36. (b) Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11, P(7 or 11). (c) Given that 8 is the outcome on the first roll, find the probability that the bettor now rolls the point 8 before rolling a 7 and thus wins. Note that at this stage in the game the only outcomes of interest are 7 and 8. Thus find P(8 | 7 or 8). (d) The probability that a bettor rolls an 8 on the first roll and then wins is given by P(8)P(8 | 7 or 8). Show that this probability is (5/36)(5/11). (e) Show that the total probability that a bettor wins in the game of craps is 0.49293. Hint: Note that the bettor can win in one of several mutually exclusive ways: by rolling a 7 or an 11 on the first roll or by establishing one of the points 4, 5, 6, 8, 9, or 10 on the first roll and then obtaining that point on successive rolls before a 7 comes up.

Paper is often tested for burst strength and tear strength. Say we classify these strengths as low, middle, and high. Then, after examining 100 pieces of paper, we find the following:Burst Strength Tear Strength A1 (low) A2 (middle) A3 (high) B1 (low) 7 11 13 B2 (middle) 11 21 9 B3 (high) 12 9 7 If we select one of the pieces at random, what are the probabilities that it has the following characteristics: (a) A1? (b) A3 B2? (c) A2 B3? (d) A1, given that it is B2? (e) B1, given that it is A3?

An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is 151/300. How many red balls are in the second urn?

Bowl A contains three red and two white chips, and bowl B contains four red and three white chips. A chip is drawn at random from bowl A and transferred to bowl B. Compute the probability of then drawing a red chip from bowl B.