Problem 43 Problem 50
A list contains 1968 statements, numbered in serial order. For each k, the kth If two marbles are removed at random from a bag containing lack and white
statement is: “This list contains exactly k false statements.” Determine the marbles, the chance that they are both white is 1/3. If 3 are removed at
truth or falsity of each statement. random, the chance that they all are white is 1/6. How many marbles are
there of each color?
Ans: Statement 1967 is true and all the rest are false
Ans: 6 white, 4 black
Problem 44
Hard Knox College is a member of a six-school basketball league in which Problem 51
every pair of schools plays each other twice. The other five schools ended Rigorously speaking, two men are “brothers-in-law” if one is married to the
the season with respective league records of .200, .300, .500, .600, and full sister of the other. How many men can there be with each man a brother-
.800. How did Hard Knox make out? in-law of every other man?
Ans: He tied for 2nd place with 0.600 Ans: 3 men
Problem 45 Problem 52
A gambler devised a game to be played with a friend. He bet ½ the money in Rigorously speaking, two men are “brothers-in-law” if one is married to the
his pocket on the toss of a coin; heads he won, tails he lost. The coin was full sister of the other. How many men can there be with each man a brother-
tossed and the money handed over. The offer was repeated and the game in-law of every other man?
continued. Each time the bet was for ½ the money then in his possession.
Ans: 3
Eventually the number of times he lost was equal to the number of times he
won. Quickly now! Did he gain, lose, or break even?
Ans: He lost Problem 53
A salesman visits ten cities arranged in the form of a circle, spending a day
in each. He proceeds clockwise from one city to the next, except whenever
Problem 46
leaving the tenth city. How many days must elapse before his location is
A prisoner is given 10 white balls, 10 black balls and two boxes. He is told
completely indeterminate, i.e., when he could be in any one of the ten cities?
that an executioner will draw one ball from one of the two boxes. If it is white,
the prisoner will go free; if it is black, he will die. How should the prisoner Ans: 83
arrange the balls in the boxes to give himself the best chance for survival?
Ans: 1 white ball in one box. Problem 54
Remaining balls(9 white and 10 black) in the other box. Three dart players threw simultaneously at a tic-tac-toe board, each hitting a
Chance of survival is 73.7% different square. What is the probabilit that the three hits constituted a win at
tic-tac-toe?
Problem 47 Ans: 2/21
On a certain day, our parking lot contains 999 cars, no two of which have the
same 3 digit license number. After 5:00 p.m. what is the probability that the
license numbers of the first 4 cars to leave the parking lot are in increasing Problem 55
order of magnitude? Four swimming pool builders submit sealed bids to a homeowner who is
required by law to accept the last bid that he sees, i.e., once he looks at a
Ans: 4! or 24 possible permutations of 4 cars bid, he automatically rejects all previous bids. He is not required to open all
the envelopes, of course. Assuming that all four bids are different, what
procedure will maximize his chances of accepting the lowest bid, and what
Problem 48 will be the probability of doing so?
A hospital nursery contains only two baby boys; the girls have not yet been
counted. At 2:00 p.m. a new baby is added to the nursery. A baby is then Ans: Probability of accepting the lowest bid is 11/24 if he uses
selected at random to be the first to have its footprint taken. It turns out to be the lower of the first two bids as his standard then his chance is
a boy. What is the probability that the last addition to the nursery was a girl? reduced to 10/24
Ans: 2/5
Problem 56
All the members of a fraternity play basketball while all but one play ice
Problem 49
hockey; yet the number of possible basketball teams (5 members) is the
Assume that a single depth cahrge has a probability of ½ of sinking a
same as the number of possible ice hockey teams (6 members). Assuming
submarine, ¼ of damage and ¼ of missing. Assume also that two damaging
there are enough members to form either type of team, how many are in the
explosions sink the sub. What is the probabilit that 4 depth charges will sink
fraternity?
the sub?
Ans: 15
Ans: 251/256
A list contains 1968 statements, numbered in serial order. For each k, the kth If two marbles are removed at random from a bag containing lack and white
statement is: “This list contains exactly k false statements.” Determine the marbles, the chance that they are both white is 1/3. If 3 are removed at
truth or falsity of each statement. random, the chance that they all are white is 1/6. How many marbles are
there of each color?
Ans: Statement 1967 is true and all the rest are false
Ans: 6 white, 4 black
Problem 44
Hard Knox College is a member of a six-school basketball league in which Problem 51
every pair of schools plays each other twice. The other five schools ended Rigorously speaking, two men are “brothers-in-law” if one is married to the
the season with respective league records of .200, .300, .500, .600, and full sister of the other. How many men can there be with each man a brother-
.800. How did Hard Knox make out? in-law of every other man?
Ans: He tied for 2nd place with 0.600 Ans: 3 men
Problem 45 Problem 52
A gambler devised a game to be played with a friend. He bet ½ the money in Rigorously speaking, two men are “brothers-in-law” if one is married to the
his pocket on the toss of a coin; heads he won, tails he lost. The coin was full sister of the other. How many men can there be with each man a brother-
tossed and the money handed over. The offer was repeated and the game in-law of every other man?
continued. Each time the bet was for ½ the money then in his possession.
Ans: 3
Eventually the number of times he lost was equal to the number of times he
won. Quickly now! Did he gain, lose, or break even?
Ans: He lost Problem 53
A salesman visits ten cities arranged in the form of a circle, spending a day
in each. He proceeds clockwise from one city to the next, except whenever
Problem 46
leaving the tenth city. How many days must elapse before his location is
A prisoner is given 10 white balls, 10 black balls and two boxes. He is told
completely indeterminate, i.e., when he could be in any one of the ten cities?
that an executioner will draw one ball from one of the two boxes. If it is white,
the prisoner will go free; if it is black, he will die. How should the prisoner Ans: 83
arrange the balls in the boxes to give himself the best chance for survival?
Ans: 1 white ball in one box. Problem 54
Remaining balls(9 white and 10 black) in the other box. Three dart players threw simultaneously at a tic-tac-toe board, each hitting a
Chance of survival is 73.7% different square. What is the probabilit that the three hits constituted a win at
tic-tac-toe?
Problem 47 Ans: 2/21
On a certain day, our parking lot contains 999 cars, no two of which have the
same 3 digit license number. After 5:00 p.m. what is the probability that the
license numbers of the first 4 cars to leave the parking lot are in increasing Problem 55
order of magnitude? Four swimming pool builders submit sealed bids to a homeowner who is
required by law to accept the last bid that he sees, i.e., once he looks at a
Ans: 4! or 24 possible permutations of 4 cars bid, he automatically rejects all previous bids. He is not required to open all
the envelopes, of course. Assuming that all four bids are different, what
procedure will maximize his chances of accepting the lowest bid, and what
Problem 48 will be the probability of doing so?
A hospital nursery contains only two baby boys; the girls have not yet been
counted. At 2:00 p.m. a new baby is added to the nursery. A baby is then Ans: Probability of accepting the lowest bid is 11/24 if he uses
selected at random to be the first to have its footprint taken. It turns out to be the lower of the first two bids as his standard then his chance is
a boy. What is the probability that the last addition to the nursery was a girl? reduced to 10/24
Ans: 2/5
Problem 56
All the members of a fraternity play basketball while all but one play ice
Problem 49
hockey; yet the number of possible basketball teams (5 members) is the
Assume that a single depth cahrge has a probability of ½ of sinking a
same as the number of possible ice hockey teams (6 members). Assuming
submarine, ¼ of damage and ¼ of missing. Assume also that two damaging
there are enough members to form either type of team, how many are in the
explosions sink the sub. What is the probabilit that 4 depth charges will sink
fraternity?
the sub?
Ans: 15
Ans: 251/256
