Probability
問題一覧
1
On the average, how many times must a die be thrown until one gets a 6?
6
2
In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does
not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win ?
1/16
3
We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?
4 x 13!39!/52!
4
The game of craps, played with two dice, is one of America's fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.
The rules are these. Only totals for the two dice count The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 1 2. Any other throw is called his "point." If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player's chance to win?
0.49293
5
Suppose King Arthur holds a jousting tournament where the jousts are in pairs as in a tennis tournament. The 8 knights in the tournament are evenly matched, and they include the twin knights Balin and Balan. What is the chance that the twins meet in a match during the tournament?
1/4
6
A plant has three suppliers. S₁ supplies 30% of the parts of the plant, S₂ supplies 50% of the parts, and S₃ supplies the remaining 20%. One percent of the parts supplied by S₁ are defective, two percent of the parts supplied by S₂ are defective, and three parts supplied by S₃ are defective. Given that the part was defective, what is the probability that it came from supplier S₃?
0.3158
7
Suppose 3 items are inspected and if at least one defective is found, the lot will be 100% inspected. Otherwise, the lot will be passed on. How likely is it that a lot containing 5 defectives will be passed on?
0.855999
8
As a project, a freshmen communications major is assigned the following project: Randomly select 125 students and count how many are talking on a cell phone at a randomly chosen time. The major counts 83 students that are talking on a cell phone. What is the relative frequency probability that a student randomly selected at a random time will be talking on a cell phone?
0.664
9
In a group of 150 graduate engineering students, 90 are enrolled in an advanced course in statistics, 60 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either course?
40
10
A manufacturer estimates that 0.25% of his output of a component are defective. The components are marketed in packets of 200. Using Poisson’s distribution, determine the probability of a packet containing only 2 defective components.
0.0758
11
Coupons in cereal boxes are numbered I to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set?
11.42
12
Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?
7 7/15
13
When 100 coins are tossed, what is the probability that exactly 50 are heads?
0.07959
14
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 foots the bill). what is the expected number of throws needed in order to determine a loser?
1½
15
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
16
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
17
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).
about .168
18
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')
0
19
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?
0.486
20
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
21
In the final seconds of the game, your favorite N.B.A. 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 (about 69%).
22
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?
one part in 750
23
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?
432516
24
A game is played as follows. You pay $1 to play. A coin is flipped four times. If four tails or four heads are obtained, you get your $1 back plus $5 more. Otherwise you forfeit you $1. What is the mathematical expectation?
-$0.25
25
An engineering company prepares an estimate for a job. The cost of preparing the estimate is $10,000. The amount of profit over and above the $10,000 is $25,000 if their estimate is accepted. The probability that their estimate will be accepted is 0.7 and the probability that their estimate will not be accepted is 0.3. What is the expected profit?
$14,500
26
A plant has three suppliers. S₁ supplies 30% of the parts of the plant, S₂ supplies 50% of the parts, and S₃ supplies the remaining 20%. One percent of the parts supplied by S₁ are defective, two percent of the parts supplied by S₂ are defective, and three parts supplied by S₃ are defective. Given that the part was defective, what is the probability that it came from supplier S₃?
0.3158
27
An engineer selects a sample of 5 iPods from a shipment of 100 that contains 5 defectives. Find the probability that the sample contains at least one defective.
0.23
28
Suppose 3 items are inspected and if at least one defective is found, the lot will be 100% inspected. Otherwise, the lot will be passed on. How likely is it that a lot containing 5 defectives will be passed on?
0.855999
29
As a project, a freshmen communications major is assigned the following project: Randomly select 125 students and count how many are talking on a cell phone at a randomly chosen time. The major counts 83 students that are talking on a cell phone. What is the relative frequency probability that a student randomly selected at a random time will be talking on a cell phone?n
0.664
30
A box of bolts contains 4 thick bolts, 3 medium bolts and 3 thin bolts. A box of nuts contains 5 that fit thick bolts, 5 that fit medium bolts, and 5 that fit thin bolts. One bolt and one nut are chosen at random. What is the classical probability that the nut fits the bolt?
1/3
31
A Ford has engines in three sizes. Of the Ford cars sold, 50% have the smallest engine, 40% have the medium engine. Of the cars with the smallest engine, 15% fail an emissions test within two years of purchase. The failure figure for medium size engines is 10%, and the failure figure for the largest engines is 5%. What is the probability that this Ford will fail the emissions test within two years?
0.12
32
In a group of 150 graduate engineering students, 90 are enrolled in an advanced course in statistics, 60 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either course?
40
33
In an unbiased coin toss, the probability of getting heads or tails is exactly ½. A coin is tossed and one gets heads. If the coin is tossed again, the probability of getting head again is:
1/2
34
The probability that a person who undergoes a kidney operation will recover is 0.6. Find the probability that of the six patients who undergo similar operations:
(a) none will recover
(b) all will recover
(c) half will recover
(d) at least half will recover
0.0041, 0.0467, 0.2765, 0.8203
35
In a used car lot, suppose 50 percent of the cars are manufactured in the United States and 15 percent of these are compact; 30 percent of the cars are manufactured in Europe and 40 percent of these are compact; and, finally, 20 percent are manufactured in Japan and 60 percent of these are compact. If a car is picked at random from the lot:
(a) find the probability that it is a compact
(b) given that the car is a compact, find the probability that it is European.
0.315, 8/21
36
A machine has fourteen identical components that function independently. It will stop working if three or more components fail. If the probability that a component fails is equal to 0.1, find the probability that the machine will be working.
0.842
37
A mouse is inside a room and each of the four walls of the room has a door through which the mouse could attempt to escape. Unluckily for the mouse, there is a trap at each of the doors d1, d2, d3, d4, and they work with probabilities 0.3, 0.2, 0.3, and 0.5, respectively. If the mouse picks a door at random, given that the mouse escapes, what is the probability that the mouse chose door d3 to escape?
0.259
38
A drug manufacturing company is debating whether a vaccine is safe enough to be marketed. The company claims that the vaccine is 90 percent effective; that is, when tried on a person, the chance for that person to develop immunity is 0.9. The federal drug agency, however, believes that the claim is exaggerated and that the drug is 40 percent effective. To test the company claim, the following procedure is devised: The vaccine will be tried on ten people. If eight or more people develop immunity, the company claim will be granted. Find the probability that:
(a) the company claim will be granted incorrectly, that is, the company claim will be granted when the federal drug agency is correct in its assertion
(b) the company claim will be denied incorrectly, that is, the company claim will be denied when the vaccine is 90 percent effective.
0.013, 0.069
39
Mrs. Crandall has owned five stocks for a number of years. She has concluded during the years she has owned the stocks that the probabilities that exactly 1, 2, 3, 4, or 5 of her stocks go up in price during a trading session are, respectively, 0.18, 0.30, 0.20, 0.15, and 0.08. Let X represent the number of Mrs. Crandall's stocks that go up in price on a trading session. Find the probability that at most two stocks go up during a trading session.
0.57
40
A test consists of five questions, and to pass the test a student has to answer at least four questions correctly. Each question has three possible answers, of which only one is correct. If a student guesses on each question, what is the probability that the student will pass the test?
less than 5 in 100
41
An individual is picked at random from a group of fifty-two athletes. Suppose twenty-six of the athletes are female and six of them are swimmers. Also, there are ten swimmers among the males. Given that the individual picked is a female, find the probability that she is a swimmer.
3/13
42
A genetic pool contains 80 alleles of which 32 are of Type A and 48 are of Type a. Suppose 15 alleles are picked at random. Find the probability of getting four A alleles, the mean number of A alleles, and the standard deviation of the number of A alleles if the sample is picked
(a) with replacement
(b) without replacement
(a) 0.127, 6, 1.9 (b) 0.122, 6, 1.72
43
In a raffle the prizes include one first prize of $3000, five second prizes of $1000 each, and twenty third prizes of $100 each. In all, 10,000 tickets are sold at $ 1 .50 each. What are the expected net winnings of a person who buys one ticket?
-0.5
44
In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does
not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win ?
1/16
45
We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?
4 x 13!39!/52!
46
Coupons in cereal boxes are numbered I to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set ?
11.42
47
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 codes
48
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')
0
49
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
50
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?
one part in 750
51
On the average, how many times must a die be thrown until one gets a 6 ?
6
52
Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one of the numbers I, 2, 3, 4, 5, 6. Three dice are rolled. If the player's number appears on one, two, or three of the dice, he
receives respectively one, two, or three times his original stake plus his own money back; otherwise he loses his stake. What is the player's expected loss per unit stake? (Actually the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)
about 8% per play
53
The game of craps, played with two dice, is one of America's fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.
The rules are these. Only totals for the two dice count The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 1 2. Any other throw is called his "point." If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player's chance to win?
0.49293
54
Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?
7 7/15
55
When 100 coins are tossed, what is the probability that exactly 50 are heads?
0.07959
56
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
57
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).
about .168
58
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?
0.486
59
In the final seconds of the game, your favorite N.B.A. 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 (about 69%)
60
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term
432516
61
A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2? How small if the number of black socks is even?
4, 21
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6問 • 1年前問題一覧
1
On the average, how many times must a die be thrown until one gets a 6?
6
2
In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does
not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win ?
1/16
3
We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?
4 x 13!39!/52!
4
The game of craps, played with two dice, is one of America's fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.
The rules are these. Only totals for the two dice count The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 1 2. Any other throw is called his "point." If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player's chance to win?
0.49293
5
Suppose King Arthur holds a jousting tournament where the jousts are in pairs as in a tennis tournament. The 8 knights in the tournament are evenly matched, and they include the twin knights Balin and Balan. What is the chance that the twins meet in a match during the tournament?
1/4
6
A plant has three suppliers. S₁ supplies 30% of the parts of the plant, S₂ supplies 50% of the parts, and S₃ supplies the remaining 20%. One percent of the parts supplied by S₁ are defective, two percent of the parts supplied by S₂ are defective, and three parts supplied by S₃ are defective. Given that the part was defective, what is the probability that it came from supplier S₃?
0.3158
7
Suppose 3 items are inspected and if at least one defective is found, the lot will be 100% inspected. Otherwise, the lot will be passed on. How likely is it that a lot containing 5 defectives will be passed on?
0.855999
8
As a project, a freshmen communications major is assigned the following project: Randomly select 125 students and count how many are talking on a cell phone at a randomly chosen time. The major counts 83 students that are talking on a cell phone. What is the relative frequency probability that a student randomly selected at a random time will be talking on a cell phone?
0.664
9
In a group of 150 graduate engineering students, 90 are enrolled in an advanced course in statistics, 60 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either course?
40
10
A manufacturer estimates that 0.25% of his output of a component are defective. The components are marketed in packets of 200. Using Poisson’s distribution, determine the probability of a packet containing only 2 defective components.
0.0758
11
Coupons in cereal boxes are numbered I to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set?
11.42
12
Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?
7 7/15
13
When 100 coins are tossed, what is the probability that exactly 50 are heads?
0.07959
14
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 foots the bill). what is the expected number of throws needed in order to determine a loser?
1½
15
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
16
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
17
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).
about .168
18
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')
0
19
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?
0.486
20
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
21
In the final seconds of the game, your favorite N.B.A. 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 (about 69%).
22
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?
one part in 750
23
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?
432516
24
A game is played as follows. You pay $1 to play. A coin is flipped four times. If four tails or four heads are obtained, you get your $1 back plus $5 more. Otherwise you forfeit you $1. What is the mathematical expectation?
-$0.25
25
An engineering company prepares an estimate for a job. The cost of preparing the estimate is $10,000. The amount of profit over and above the $10,000 is $25,000 if their estimate is accepted. The probability that their estimate will be accepted is 0.7 and the probability that their estimate will not be accepted is 0.3. What is the expected profit?
$14,500
26
A plant has three suppliers. S₁ supplies 30% of the parts of the plant, S₂ supplies 50% of the parts, and S₃ supplies the remaining 20%. One percent of the parts supplied by S₁ are defective, two percent of the parts supplied by S₂ are defective, and three parts supplied by S₃ are defective. Given that the part was defective, what is the probability that it came from supplier S₃?
0.3158
27
An engineer selects a sample of 5 iPods from a shipment of 100 that contains 5 defectives. Find the probability that the sample contains at least one defective.
0.23
28
Suppose 3 items are inspected and if at least one defective is found, the lot will be 100% inspected. Otherwise, the lot will be passed on. How likely is it that a lot containing 5 defectives will be passed on?
0.855999
29
As a project, a freshmen communications major is assigned the following project: Randomly select 125 students and count how many are talking on a cell phone at a randomly chosen time. The major counts 83 students that are talking on a cell phone. What is the relative frequency probability that a student randomly selected at a random time will be talking on a cell phone?n
0.664
30
A box of bolts contains 4 thick bolts, 3 medium bolts and 3 thin bolts. A box of nuts contains 5 that fit thick bolts, 5 that fit medium bolts, and 5 that fit thin bolts. One bolt and one nut are chosen at random. What is the classical probability that the nut fits the bolt?
1/3
31
A Ford has engines in three sizes. Of the Ford cars sold, 50% have the smallest engine, 40% have the medium engine. Of the cars with the smallest engine, 15% fail an emissions test within two years of purchase. The failure figure for medium size engines is 10%, and the failure figure for the largest engines is 5%. What is the probability that this Ford will fail the emissions test within two years?
0.12
32
In a group of 150 graduate engineering students, 90 are enrolled in an advanced course in statistics, 60 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either course?
40
33
In an unbiased coin toss, the probability of getting heads or tails is exactly ½. A coin is tossed and one gets heads. If the coin is tossed again, the probability of getting head again is:
1/2
34
The probability that a person who undergoes a kidney operation will recover is 0.6. Find the probability that of the six patients who undergo similar operations:
(a) none will recover
(b) all will recover
(c) half will recover
(d) at least half will recover
0.0041, 0.0467, 0.2765, 0.8203
35
In a used car lot, suppose 50 percent of the cars are manufactured in the United States and 15 percent of these are compact; 30 percent of the cars are manufactured in Europe and 40 percent of these are compact; and, finally, 20 percent are manufactured in Japan and 60 percent of these are compact. If a car is picked at random from the lot:
(a) find the probability that it is a compact
(b) given that the car is a compact, find the probability that it is European.
0.315, 8/21
36
A machine has fourteen identical components that function independently. It will stop working if three or more components fail. If the probability that a component fails is equal to 0.1, find the probability that the machine will be working.
0.842
37
A mouse is inside a room and each of the four walls of the room has a door through which the mouse could attempt to escape. Unluckily for the mouse, there is a trap at each of the doors d1, d2, d3, d4, and they work with probabilities 0.3, 0.2, 0.3, and 0.5, respectively. If the mouse picks a door at random, given that the mouse escapes, what is the probability that the mouse chose door d3 to escape?
0.259
38
A drug manufacturing company is debating whether a vaccine is safe enough to be marketed. The company claims that the vaccine is 90 percent effective; that is, when tried on a person, the chance for that person to develop immunity is 0.9. The federal drug agency, however, believes that the claim is exaggerated and that the drug is 40 percent effective. To test the company claim, the following procedure is devised: The vaccine will be tried on ten people. If eight or more people develop immunity, the company claim will be granted. Find the probability that:
(a) the company claim will be granted incorrectly, that is, the company claim will be granted when the federal drug agency is correct in its assertion
(b) the company claim will be denied incorrectly, that is, the company claim will be denied when the vaccine is 90 percent effective.
0.013, 0.069
39
Mrs. Crandall has owned five stocks for a number of years. She has concluded during the years she has owned the stocks that the probabilities that exactly 1, 2, 3, 4, or 5 of her stocks go up in price during a trading session are, respectively, 0.18, 0.30, 0.20, 0.15, and 0.08. Let X represent the number of Mrs. Crandall's stocks that go up in price on a trading session. Find the probability that at most two stocks go up during a trading session.
0.57
40
A test consists of five questions, and to pass the test a student has to answer at least four questions correctly. Each question has three possible answers, of which only one is correct. If a student guesses on each question, what is the probability that the student will pass the test?
less than 5 in 100
41
An individual is picked at random from a group of fifty-two athletes. Suppose twenty-six of the athletes are female and six of them are swimmers. Also, there are ten swimmers among the males. Given that the individual picked is a female, find the probability that she is a swimmer.
3/13
42
A genetic pool contains 80 alleles of which 32 are of Type A and 48 are of Type a. Suppose 15 alleles are picked at random. Find the probability of getting four A alleles, the mean number of A alleles, and the standard deviation of the number of A alleles if the sample is picked
(a) with replacement
(b) without replacement
(a) 0.127, 6, 1.9 (b) 0.122, 6, 1.72
43
In a raffle the prizes include one first prize of $3000, five second prizes of $1000 each, and twenty third prizes of $100 each. In all, 10,000 tickets are sold at $ 1 .50 each. What are the expected net winnings of a person who buys one ticket?
-0.5
44
In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does
not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win ?
1/16
45
We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?
4 x 13!39!/52!
46
Coupons in cereal boxes are numbered I to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set ?
11.42
47
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 codes
48
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')
0
49
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
50
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?
one part in 750
51
On the average, how many times must a die be thrown until one gets a 6 ?
6
52
Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one of the numbers I, 2, 3, 4, 5, 6. Three dice are rolled. If the player's number appears on one, two, or three of the dice, he
receives respectively one, two, or three times his original stake plus his own money back; otherwise he loses his stake. What is the player's expected loss per unit stake? (Actually the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)
about 8% per play
53
The game of craps, played with two dice, is one of America's fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.
The rules are these. Only totals for the two dice count The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 1 2. Any other throw is called his "point." If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player's chance to win?
0.49293
54
Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?
7 7/15
55
When 100 coins are tossed, what is the probability that exactly 50 are heads?
0.07959
56
There are three families, each with two sons and two daughters. In how many ways can all these young people be married?
80
57
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).
about .168
58
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?
0.486
59
In the final seconds of the game, your favorite N.B.A. 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 (about 69%)
60
If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term
432516
61
A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2? How small if the number of black socks is even?
4, 21