Question 1

Solve the problem.
-If 2 cards are drawn from a deck of 52 cards, what is the expected number of spades?
A) 0.25
B) 0.47
C) 0.50
D) 0.75

Accepted Answer

The probability of drawing a spade from a deck of 52 cards is 1/4, as there are 13 spades in a deck. For 2 cards, the expected number of spades is 2 * (1/4) = 0.50.

Question 2

Solve the problem.
-Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for Rolling a 6 or a 5, nothing otherwise. What is your expected net winnings?
A) $0.00
B) -$1.00
C) $1.00
D) $3.00

Accepted Answer

The expected net winnings are calculated by multiplying each outcome by its probability and summing these products. Rolling a 5 or 6 (2 outcomes out of 6) gives $2 net gain ($3 received minus $1 paid), and the other 4 outcomes result in a $1 loss (nothing received minus $1 paid). The expected net winnings = (2/6)*$2 + (4/6)*(-$1) = $0.

Question 3

Solve the problem.
-Numbers is a game where you bet $1.00 on any three-digit number from 000 to 999. If your Number comes up, you get $600.00. Find the expected winnings.
A) -$0.42
B) -$0.40
C) -$0.50
D) -$1.00

Accepted Answer

The answer of Solve the problem.
-Numbers is a game where...

Question 4

Solve the problem.
-Find the expected number of girls in a family of 5 children.
A) 2.25
B) 2.5
C) 3
D) 2

Accepted Answer

The expected number of girls in a family of 5 children can be calculated assuming the probability of having a girl is 0.5 (or 50%) for each child. The expected value is calculated as the number of trials (children) multiplied by the probability of success (having a girl), which is 5 * 0.5 = 2.5.

Question 5

Solve the problem.
-From a group of 3 men and 4 women, a delegation of 2 is selected. What is the expected number of Men in the delegation?
A) 0.48
B) 1
C) 0.86
D) 0.57

Accepted Answer

The expected number of men is calculated by finding the probability of selecting 0, 1, or 2 men and multiplying by the number of men, then summing these products. The probability of selecting 0 men is (4C2)/(7C2), 1 man is (3C1 * 4C1)/(7C2), and 2 men is (3C2)/(7C2). The expected value is 0*(probability of 0 men) + 1*(probability of 1 man) + 2*(probability of 2 men) = 0.86.

Question 6

Solve the problem.
-Experience shows that a ski lodge will be full (168 guests) if there is a heavy snow fall in December, while only partially full (74 guests) with a light snow fall. What is the expected number Of guests if the probability for a heavy snow fall is 0.40?
A) 111.6
B) 67.2
C) 100.8
D) 130.4

Accepted Answer

The expected number of guests can be calculated using the formula for expected value: E(X) = (probability of heavy snowfall * number of guests with heavy snowfall) + (probability of light snowfall * number of guests with light snowfall). Given the probability of heavy snowfall is 0.40, the probability of light snowfall is 1 - 0.40 = 0.60. Thus, E(X) = (0.40 * 168) + (0.60 * 74) = 67.2 + 44.4 = 111.6.

Question 7

Solve the problem.
-Bob and Fred play the following game. Bob rolls a single die. If an even number results, Bob must Pay Fred the number of dollars indicated by the number rolled. On the other hand, if an odd Number is rolled, Fred must pay Bob the number of dollars indicated by the number rolled. Find Fredʹs expectation.
A) $0.00
B) -$0.10
C) $0.50
D) $0.25

Accepted Answer

The answer of Solve the problem.
-Bob and Fred play the...

Question 8

Solve the problem.
-Find the expected number of girls in a family of 3 children.
A) 2
B) 1.5
C) 2.25
D) 2.5

Accepted Answer

The expected number of girls in a family of 3 children can be calculated by multiplying the probability of having a girl (1/2) by the number of children (3). So, the expected number of girls is (1/2) * 3 = 1.5.

Question 9

Solve the problem.
-Bob and Fred play the following game. Bob rolls a single die. If an even number results, Bob must Pay Fred the number of dollars indicated by the number rolled. On the other hand, if an odd Number is rolled, Fred must pay Bob the number of dollars indicated by the number rolled. Find Bobʹs expectation.
A) $0.00
B) $0.40
C) -$0.50
D) -$0.25

Accepted Answer

The answer of Solve the problem.
-Bob and Fred play the...

Question 10

Solve the problem.
-Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $6.00 for Rolling a 6 or a 2, nothing otherwise. What are your expected winnings?
A) $6.00
B) -$2.00
C) $0.00
D) $2.00

Accepted Answer

The probability of rolling a 6 or a 2 is 2/6 (or 1/3), since there are two favorable outcomes out of six possible outcomes. The expected winnings for each roll are calculated as follows: (1/3) * $6.00 + (2/3) * (-$2.00) = $2.00 - $4.00/3 = $0.00. This means, on average, you neither gain nor lose money.

Question 11

Solve the problem.
-Bob and Fred play the following game. Bob rolls a single die. If the result is the number 1, 2, 3, 4, Or 5 , Bob must pay Fred the number of dollars indicated by the number rolled. If a 6 is rolled, Fred must pay Bob $45. Find Fredʹs expectation.
A) $4.00
B) $0.00
C) -$10.00
D) -$5.00

Accepted Answer

Fred's expectation can be calculated by taking the average of all possible outcomes. For rolls 1 through 5, Fred gains $1, $2, $3, $4, and $5 respectively, which sums to $15. For a roll of 6, Fred loses $45. The expectation is the sum of these outcomes divided by the number of outcomes: (1+2+3+4+5-45)/6 = -30/6 = -$5.

Question 12

Solve the problem.
-Suppose a charitable organization decides to raise money by raffling a trip worth $500. If 3,000 Tickets are sold at $1.00 each, find the expected value of winning for a person who buys 1 ticket.
A) -$0.81
B) -$0.85
C) -$0.83
D) -$1.00

Accepted Answer

The answer of Solve the problem.
-Suppose a charitable organization decides...

Question 13

Solve the problem.
-An insurance company says that at age 50 one must choose to take $10,000 at age 60, $30,000 at 70, Or $50,000 at 80 ($0 death benefit). The probability of living from 50 to 60 is 0.87, from 50 to 70, 0.62, and from 50 to 80, 0.42. Find the expected value at each age. 
A) $ 60: \$ 8700 $
$ 70: \$ 12,600 $
$ 80: \$ 21,000 $
 
B) $ 60: \$ 6200 $
$ 70: \$ 18,600 $
$ 80: \$ 21,000 $

C) 60: $ \$ 8700 $
70: $ \$ 18,600 $
80: \$21,000

D) 60: $ \$ 8700 $
$ 70: \$ 26,100 $
80: $ \$ 43,500 $

Accepted Answer

The expected value at each age is calculated by multiplying the probability of living to that age by the amount of money received at that age. For age 60, it's $0.87 \times \$10,000 = \$8,700$. For age 70, it's $0.62 \times \$30,000 = \$18,600$. For age 80, it's $0.42 \times \$50,000 = \$21,000$.

Question 14

Solve the problem.
-At age 50, Ann must choose between taking $18,000 at age 60 if she is alive then, or $30,000 at age 70 if she is alive then. The probability for a person aged 50 living to be 60 and 70 is 0.81 and 0.61, Respectively. Using expected value, what is Annʹs best option?
A) $30,000 at age 70
B) $18,000 at age 60

Accepted Answer

The expected value of taking $18,000 at age 60 is 0.81 * $18,000 = $14,580. The expected value of taking $30,000 at age 70 is 0.61 * $30,000 = $18,300. Therefore, Ann's best option is to take $30,000 at age 70.

Question 15

Solve the problem.
-An insurance company has written 62 policies of $50,000, 455 of $25,000, and 908 of $10,000 on People of age 20. If the probability that a person will die at age 20 is 0.001, how much can the Company expect to pay during the year the policies were written?
A) $2356
B) $23,555
C) $235,550
D) $0

Accepted Answer

The expected payout for each policy is calculated by multiplying the payout amount by the probability of the event (death at age 20) occurring. Therefore, for each policy type, the expected payout is:- For $50,000 policies: $50,000 * 0.001 = $50 per policy. With 62 policies, this totals $50 * 62 = $3,100.- For $25,000 policies: $25,000 * 0.001 = $25 per policy. With 455 policies, this totals $25 * 455 = $11,375.- For $10,000 policies: $10,000 * 0.001 = $10 per policy. With 908 policies, this totals $10 * 908 = $9,080.Adding these totals together gives the expected payout for the year: $3,100 + $11,375 + $9,080 = $23,555.

Question 16

Solve the problem.
-In a game, you have a 1/21 probability of winning $107 and a 20/21 probability of losing $5. What Is your expected winning?
A) $5.10
B) $9.86
C) $0.33
D) -$4.76

Accepted Answer

The expected winning is calculated by multiplying each outcome by its probability and summing these products. For winning $107, the expectation is (1/21) * $107 = $5.10. For losing $5, the expectation is (20/21) * -$5 = -$4.76. Adding these gives $5.10 - $4.76 = $0.33.

Question 17

Solve the problem.
-If 3 balls are drawn from a bag containing 3 red and 4 blue balls, what is the expected number of Red balls in the sample?
A) 1.29
B) 0.89
C) 1.39
D) 1.54

Accepted Answer

The expected number of Red balls can be calculated using the formula for expectation in a hypergeometric distribution, which is $E[X] = n \times \frac{K}{N}$, where $n$ is the number of draws (3 balls), $K$ is the total number of Red balls (3), and $N$ is the total number of balls (7). Thus, $E[X] = 3 \times \frac{3}{7} = \frac{9}{7} \approx 1.29$.

Question 18

Solve the problem.
-A contractor is considering a sale that promises a profit of $26,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $1000 with a probability of 0.3. What is the expected Profit?
A) $18,200
B) $25,000
C) $18,900
D) $17,900

Accepted Answer

The answer of Solve the problem.
-A contractor is considering a...

Question 19

Solve the problem.
-If 5 apples in a barrel of 25 apples are rotten, what is the expected number of rotten apples in a Sample of 2 apples?
A) 0.63
B) 1
C) 0.4
D) 0.33

Accepted Answer

The expected number of rotten apples in a sample of 2 apples can be calculated using the formula for expectation in probability. With 5 rotten apples out of 25, the probability of picking a rotten apple is 5/25 or 1/5. For a sample of 2 apples, the expected number of rotten apples is 2*(1/5) = 2/5 = 0.4.

Question 20

Solve the problem.
-Find the expected number of boys in a family of 6 children.
A) 3
B) 3.5
C) 3.75
D) 4

Accepted Answer

The expected number of boys in a family of 6 children is calculated assuming the probability of having a boy is 0.5 (or 50%). Therefore, the expected number of boys = 6 children * 0.5 = 3.

Quiz 11: Probability

Solve the problem. -If 2 cards are drawn from a deck of 52 cards, what is the expected number of spades?

Solve the problem. -Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for Rolling a 6 or a 5, nothing otherwise. What is your expected net winnings?

Solve the problem. -Numbers is a game where you bet $1.00 on any three-digit number from 000 to 999. If your Number comes up, you get $600.00. Find the expected winnings.

Solve the problem. -Find the expected number of girls in a family of 5 children.

Solve the problem. -From a group of 3 men and 4 women, a delegation of 2 is selected. What is the expected number of Men in the delegation?

Solve the problem. -Experience shows that a ski lodge will be full (168 guests) if there is a heavy snow fall in December, while only partially full (74 guests) with a light snow fall. What is the expected number Of guests if the probability for a heavy snow fall is 0.40?

Solve the problem. -Find the expected number of girls in a family of 3 children.

Solve the problem. -Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $6.00 for Rolling a 6 or a 2, nothing otherwise. What are your expected winnings?

Solve the problem. -Bob and Fred play the following game. Bob rolls a single die. If the result is the number 1, 2, 3, 4, Or 5 , Bob must pay Fred the number of dollars indicated by the number rolled. If a 6 is rolled, Fred must pay Bob $45. Find Fredʹs expectation.

Solve the problem. -Suppose a charitable organization decides to raise money by raffling a trip worth $500. If 3,000 Tickets are sold at $1.00 each, find the expected value of winning for a person who buys 1 ticket.

Solve the problem. -An insurance company says that at age 50 one must choose to take $10,000 at age 60, $30,000 at 70, Or $50,000 at 80 ($0 death benefit) . The probability of living from 50 to 60 is 0.87, from 50 to 70, 0.62, and from 50 to 80, 0.42. Find the expected value at each age.

Solve the problem. -At age 50, Ann must choose between taking $18,000 at age 60 if she is alive then, or $30,000 at age 70 if she is alive then. The probability for a person aged 50 living to be 60 and 70 is 0.81 and 0.61, Respectively. Using expected value, what is Annʹs best option?

Solve the problem. -An insurance company has written 62 policies of $50,000, 455 of $25,000, and 908 of $10,000 on People of age 20. If the probability that a person will die at age 20 is 0.001, how much can the Company expect to pay during the year the policies were written?

Solve the problem. -In a game, you have a 1/21 probability of winning $107 and a 20/21 probability of losing $5. What Is your expected winning?

Solve the problem. -If 3 balls are drawn from a bag containing 3 red and 4 blue balls, what is the expected number of Red balls in the sample?

Solve the problem. -A contractor is considering a sale that promises a profit of $26,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $1000 with a probability of 0.3. What is the expected Profit?

Solve the problem. -If 5 apples in a barrel of 25 apples are rotten, what is the expected number of rotten apples in a Sample of 2 apples?

Solve the problem. -Find the expected number of boys in a family of 6 children.