Mathematical Bafflers
問題一覧
1
A gambler devised a game to be played with a friend. He bet ½ the money in his pocket on the toss of a coin; heads he won, tails he lost. The coin was tossed, and the money handed over. The offer was repeated, and the game continued. Each time he bet was for ½ the money then in his possession. 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?
He lost, even if they played only twice, or four times, or six.
2
A prisoner is given 10 white balls, 10 black balls and two boxes. He is told 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 arrange the balls in the boxes to give himself the best chance for survival?
If the prisoner places one white ball in one box and the remaining balls in other box. his chance of survival would be 0.737
3
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 order of magnitude.
1/24
4
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 selected at a random to be the first to have its footprint taken. It turns out to be a boy. What is the probability that the last addition to the nursery was girl?
2/5
5
Assume that a single depth charge has a probability of ½ of sinking a submarine, ¼ of damage and ¼ of missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges will sink the sub?
251/246
6
If 2 marbles are removed at a random from a bag containing black and white marbles, the chance that they are both white is ⅓. If 3 are removed at random, the chance that they all are white is 1/6. How many marbles are there of each color?
6 white and 4 black
7
An expert gives team A only a 40% chance to win the World Series. Basing his calculation on this gambler offers 6 to 5 odds on team B to win the first game. Is his judgement sound?
the gamble is on the safe side, P =0.5455
8
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 leaving the tenth city he may go to either the first or jump to the second city. How many days must elapse before his location is completely indeterminate, i.e., when he could be in any one of the ten cities?
83
9
Three dart players threw simultaneously at a tic-tac-toe board, each hitting a different square. What is the probability that the three hits constituted a win at tic-tac-toe?
2/21
10
All the members of a fraternity play basketball while all but one play ice hockey; yet the number of possible basketball teams (5 members) is the same as the number of possible ice hockey teams (6 members). Assuming there are enough members to form either type of team, how many are in the fraternity?
15 members
11
A game of super-dominoes is played with pieces divided into three cells instead of the usual two, containing all combinations from triple blank to triple six, with no duplications. For example the set does not include both 1 2 3 and 3 2 1 since these are merely reversals of each other. (But, it does contain 1 3 2.) How many pieces are there in a set?
196
12
Martian coins are 3-sided (heads, tails, and torsos), each side coming up with equal probability. Three Martians decided to go odd-man-out to determine who pays a dinner check. (If two coins come up the same and one different, the owner of the latter coin foot the bill. What is the expected number of throws needed in order to determine a loser?
1 1/2
13
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
14
How many three digit telephone area codes are possible given that: (a) the first digit must not be zero or one; (b) the second digit must be zero or one; (c) the third digit must not be zero; (d) the third digit may be one only if the second digit is zero.
136 possible cars
15
Six men decide to play Russian roulette with a six gun loaded with one cartridge. They draw for position, and afterwards, the sixth man casually suggests that instead of letting the chamber rotate in sequence, each man spin the chamber before shooting. How would this improve his chances?
0.1
16
A long shot poker player draws two cards to the five and six of diamonds and the joker. What are his chances of coming up with a pat hand? (straight or flush)
0.168
17
In Puevigi, the game of craps is played with a referee calling the point by adding together the six faces (three on each die) visible from his vantage point. What is the probability of making 16 the hard way? (That is, by throwing two eights.
zero
18
Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either of Min's dice displays the same number of spots as either of Max's, she wins and they go to Bermuda. Otherwise, they go to Yellowstone. What is the chance they'll see "Old Faithful" this year?
0.514
19
There are four volumes of an encyclopedia on a shelf, each volume containing 300 pages, (that is, numbered 1 to 600), but these have been placed on the shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his way through to the last page of Vol. 4. What is the expected number of pages (excluding covers) he has eaten through?
500
20
Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come? With four specimens, the odds in favour of a mated triple are
4/9
21
Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come? if payload limitation permit five to travel to Earth, the odds go up to
50/81
22
In the final seconds of the game, your favorite NBA team is behind 117 to 118. Your center attempts a shot and is fouled for the 2nd time in the last 2 minutes as the buzzer sounds. Three to make two in the penalty situation. Optimistic?
Note: the center is only a 50% free-thrower. What are your team's overall chances of winning?
11/16
23
One of a pair of dice is loaded so that the chance of a 1 turning up is 1/5, the other faces being equally likely. Its mate is loaded so that the chance of a 6 turning up 1/5, the other faces being equally likely. How much does this loading increase the probability of throwing a 7 with the two dice?
750
24
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?
432516
25
The local weather forecaster says "no rain" and his record is 2/3 accuracy of prediction. But the Federal Meteorological Service predicts rain and their record is ¾4. With no other data available, what is the chance of rain?
3/5
26
A sharp operator makes the following deal. A player is to toss a coin and receive 1, 4, 9, . n? dollars if the first head comes up on the first, second, third, ... nth toss. The sucker pays ten dollars for this. How much can the operator expect to make if this is repeated a great many times?
4 dollars per game
27
In 1969, the World Series will begin in the stadium of the American League pennant winner. Assume the contenders are evenly matched. What is the probability that the series will end where it began?
5/8
28
In a carnival game 5 balls are tossed into a square box divided into 4 square cells, with baffles to ensure that every ball has an equal chance of going in any cell. The player pays $1 and receives $1 for every cell which is empty after the 5 balls are thrown. How much does the operator expect to make per game?
a nickel per game
29
Having lost a checker game, a specialist in learning programs threw one of the red checkers out the window. His wife reboxed the 12 black pieces and 11 red pieces one at a time in random fashion. The number of black checkers in the box always exceeded the number of reds. What was the a priori probability of this occurrence?
1/23
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問題一覧
1
A gambler devised a game to be played with a friend. He bet ½ the money in his pocket on the toss of a coin; heads he won, tails he lost. The coin was tossed, and the money handed over. The offer was repeated, and the game continued. Each time he bet was for ½ the money then in his possession. 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?
He lost, even if they played only twice, or four times, or six.
2
A prisoner is given 10 white balls, 10 black balls and two boxes. He is told 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 arrange the balls in the boxes to give himself the best chance for survival?
If the prisoner places one white ball in one box and the remaining balls in other box. his chance of survival would be 0.737
3
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 order of magnitude.
1/24
4
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 selected at a random to be the first to have its footprint taken. It turns out to be a boy. What is the probability that the last addition to the nursery was girl?
2/5
5
Assume that a single depth charge has a probability of ½ of sinking a submarine, ¼ of damage and ¼ of missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges will sink the sub?
251/246
6
If 2 marbles are removed at a random from a bag containing black and white marbles, the chance that they are both white is ⅓. If 3 are removed at random, the chance that they all are white is 1/6. How many marbles are there of each color?
6 white and 4 black
7
An expert gives team A only a 40% chance to win the World Series. Basing his calculation on this gambler offers 6 to 5 odds on team B to win the first game. Is his judgement sound?
the gamble is on the safe side, P =0.5455
8
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 leaving the tenth city he may go to either the first or jump to the second city. How many days must elapse before his location is completely indeterminate, i.e., when he could be in any one of the ten cities?
83
9
Three dart players threw simultaneously at a tic-tac-toe board, each hitting a different square. What is the probability that the three hits constituted a win at tic-tac-toe?
2/21
10
All the members of a fraternity play basketball while all but one play ice hockey; yet the number of possible basketball teams (5 members) is the same as the number of possible ice hockey teams (6 members). Assuming there are enough members to form either type of team, how many are in the fraternity?
15 members
11
A game of super-dominoes is played with pieces divided into three cells instead of the usual two, containing all combinations from triple blank to triple six, with no duplications. For example the set does not include both 1 2 3 and 3 2 1 since these are merely reversals of each other. (But, it does contain 1 3 2.) How many pieces are there in a set?
196
12
Martian coins are 3-sided (heads, tails, and torsos), each side coming up with equal probability. Three Martians decided to go odd-man-out to determine who pays a dinner check. (If two coins come up the same and one different, the owner of the latter coin foot the bill. What is the expected number of throws needed in order to determine a loser?
1 1/2
13
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
14
How many three digit telephone area codes are possible given that: (a) the first digit must not be zero or one; (b) the second digit must be zero or one; (c) the third digit must not be zero; (d) the third digit may be one only if the second digit is zero.
136 possible cars
15
Six men decide to play Russian roulette with a six gun loaded with one cartridge. They draw for position, and afterwards, the sixth man casually suggests that instead of letting the chamber rotate in sequence, each man spin the chamber before shooting. How would this improve his chances?
0.1
16
A long shot poker player draws two cards to the five and six of diamonds and the joker. What are his chances of coming up with a pat hand? (straight or flush)
0.168
17
In Puevigi, the game of craps is played with a referee calling the point by adding together the six faces (three on each die) visible from his vantage point. What is the probability of making 16 the hard way? (That is, by throwing two eights.
zero
18
Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either of Min's dice displays the same number of spots as either of Max's, she wins and they go to Bermuda. Otherwise, they go to Yellowstone. What is the chance they'll see "Old Faithful" this year?
0.514
19
There are four volumes of an encyclopedia on a shelf, each volume containing 300 pages, (that is, numbered 1 to 600), but these have been placed on the shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his way through to the last page of Vol. 4. What is the expected number of pages (excluding covers) he has eaten through?
500
20
Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come? With four specimens, the odds in favour of a mated triple are
4/9
21
Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come? if payload limitation permit five to travel to Earth, the odds go up to
50/81
22
In the final seconds of the game, your favorite NBA team is behind 117 to 118. Your center attempts a shot and is fouled for the 2nd time in the last 2 minutes as the buzzer sounds. Three to make two in the penalty situation. Optimistic?
Note: the center is only a 50% free-thrower. What are your team's overall chances of winning?
11/16
23
One of a pair of dice is loaded so that the chance of a 1 turning up is 1/5, the other faces being equally likely. Its mate is loaded so that the chance of a 6 turning up 1/5, the other faces being equally likely. How much does this loading increase the probability of throwing a 7 with the two dice?
750
24
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?
432516
25
The local weather forecaster says "no rain" and his record is 2/3 accuracy of prediction. But the Federal Meteorological Service predicts rain and their record is ¾4. With no other data available, what is the chance of rain?
3/5
26
A sharp operator makes the following deal. A player is to toss a coin and receive 1, 4, 9, . n? dollars if the first head comes up on the first, second, third, ... nth toss. The sucker pays ten dollars for this. How much can the operator expect to make if this is repeated a great many times?
4 dollars per game
27
In 1969, the World Series will begin in the stadium of the American League pennant winner. Assume the contenders are evenly matched. What is the probability that the series will end where it began?
5/8
28
In a carnival game 5 balls are tossed into a square box divided into 4 square cells, with baffles to ensure that every ball has an equal chance of going in any cell. The player pays $1 and receives $1 for every cell which is empty after the 5 balls are thrown. How much does the operator expect to make per game?
a nickel per game
29
Having lost a checker game, a specialist in learning programs threw one of the red checkers out the window. His wife reboxed the 12 black pieces and 11 red pieces one at a time in random fashion. The number of black checkers in the box always exceeded the number of reds. What was the a priori probability of this occurrence?
1/23