Question 1

Find the expected value.
-The prizes that can be won in a sweepstakes are listed below together with the chances of winning each one:
$5200 (1 chance in 8900); $2500 (1 chance in 6400); $900 (1 chance in 3900); $100 (1 chance in 2100).
Find the expected value of the amount won for one entry if the cost of entering is 57 cents.
A)$100.00
B)$0.62
C)$1.21
D)$0.68

Accepted Answer

The expected value (EV) is calculated by multiplying each prize amount by its probability of winning and summing these products. Subtract the cost of entering (57 cents) from the sum to find the net expected value.- For $5200, the probability is 1/8900, so EV = $5200 * (1/8900).- For $2500, the probability is 1/6400, so EV = $2500 * (1/6400).- For $900, the probability is 1/3900, so EV = $900 * (1/3900).- For $100, the probability is 1/2100, so EV = $100 * (1/2100).Summing these gives the total expected value before the cost of entering:$$EV = \frac{5200}{8900} + \frac{2500}{6400} + \frac{900}{3900} + \frac{100}{2100}$$$$EV \approx 0.58427 + 0.390625 + 0.230769 + 0.047619 = 1.25329$$Subtracting the cost of entering (57 cents or $0.57) from this sum gives the net expected value:$$Net EV = 1.25329 - 0.57 = 0.68329$$Rounded to the nearest cent, the expected value is $0.68.

Question 2

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-An insurance company reported that last year there were claims for medical expenses for 2669 on-the-job injuries. The company covers 327,171 employees. Find the rate of on-the-job injuries per 10,000 employees. Round your answer to hundredths.
A)2.669 injuries per 10,000 employees
B)81.58 injuries per 10,000 employees
C)0.01 injury per 10,000 employees
D)0.008 injury per 10,000 employees

Accepted Answer

To find the rate of on-the-job injuries per 10,000 employees, divide the number of injuries (2669) by the total number of employees (327,171) and then multiply by 10,000. This calculation gives $ \frac{2669}{327171} \times 10000 \approx 81.58 $ injuries per 10,000 employees.

Question 3

Find the expected value.
-A 28-year-old man pays $57 for a one-year life insurance policy with coverage of $130,000. If the probability that he will live through the year is 0.9994, what is the expected value for the insurance policy?
A)$21.00
B)-$56.97
C)$129,922.00
D)$78.00

Accepted Answer

The expected value is calculated by multiplying each outcome by its probability and summing these products. If the man lives, he loses the premium ($57), and if he dies, his net gain is the coverage minus the premium ($130,000 - $57 = $129,943). Thus, the expected value is (0.9994 * -$57) + (0.0006 * $129,943) = -$56.97 + $77.97 = $21.

Question 4

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-A six mile strip of Sprinkle Road had 67 traffic fatalities last year. There were 2.6 million miles driven along this stretch. Determine the traffic fatality rate per 1,000,000 miles driven. Round your answer to hundredths.
A)25.77 deaths per 1,000,000 miles driven
B)25,769,230.77 deaths per 1,000,000 miles driven
C)2.57692308 deaths per 1,000,000 miles driven
D)0.04 death per 1,000,000 miles driven

Accepted Answer

To find the traffic fatality rate per 1,000,000 miles driven, divide the number of fatalities (67) by the total miles driven (2.6 million), then multiply by 1,000,000. This gives (67 / 2,600,000) * 1,000,000 = 25.77 deaths per 1,000,000 miles driven.

Question 5

Answer the question about the Law of Large Numbers.
-A gambler playing blackjack in a casino has lost each of his first 16 bets. If he correctly understood the Law of large Numbers, which strategy would he use in betting on future blackjack hands?
A. He should bet based on the merit of each individual hand. While the proportion of wins should eventually approach
The true probability of winning, this does not mean that he is likely to recover early losses.
B. He should bet more on the next 16 hands, because the proportion of wins should quickly approach the true
Probability of winning.
B. He should bet less on the next 16 hands, because it doesn't look like this is a good day for him.
D. He shouldn't use any of the above strategies.
A)c
B)b
C)a
D)d

Accepted Answer

The Law of Large Numbers suggests that outcomes will average out to the expected probability over a large number of trials, but it does not predict short-term outcomes. Therefore, the gambler should bet based on the merit of each hand, as past losses do not influence the odds of future hands.

Question 6

Find the expected value.
-In a game, you have a $\frac { 1 } { 22 }$ probability of winning $89 and a $\frac { 21 } { 22 }$ Probability of losing $9. What is your expected
Value?
A)-$8.59
B)$12.64
C)$4.05
D)-$4.55

Accepted Answer

The expected value is calculated as: (1/22 * $89) + (21/22 * -$9) = $4.05 - $8.59 = -$4.54, which rounds to -$4.55.

Question 7

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-A six mile strip of Sprinkle Road had 50 traffic fatalities last year. There were 2.4 million miles driven along this stretch. Determine the traffic fatality rate per 100,000 miles driven. Round your answer to hundredths.
A)2.08 deaths per 100,000 miles driven
B)20.83 deaths per 100,000 miles driven
C)24 deaths per 100,000 miles driven
D)4800.00 deaths per 100,000 miles driven

Accepted Answer

To find the traffic fatality rate per 100,000 miles driven, divide the number of fatalities (50) by the total miles driven (2.4 million) and then multiply by 100,000. This gives (50 / 2,400,000) * 100,000 = 2.08 deaths per 100,000 miles driven.

Question 8

Answer the question about the Law of Large Numbers.
-John decides to play the state lotto, which requires the purchaser to select six numbers between 1 and 50. he selects 4, 4, 4, 4, 4, 4 for his numbers, believing that these are just as likely as any other series of six numbers between 1 and 50. He also notes that these numbers have not yet been selected as winning numbers. What can we accurately say about John's choice of numbers?
A. It is a reasonable choice.
B. It is an unreasonable choice, since a series of six 4s is very unlikely.
C. It is a reasonable choice, because these numbers are due to be picked.
D. It is a unreasonable choice, because these numbers have never been picked before.
A)d
B)a
C)b
D)c

Accepted Answer

The answer of Answer the question about the Law of...

Question 9

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-At an intersection in Normal, Illinois, there were 90 vehicle accidents with 168,199 vehicles passing through the intersection. Determine the accident rate per 1000 vehicles.
A)0.00054 accident per 1000 vehicles
B)0.535 accident per 1000 vehicles
C)1,868,877.78 accidents per 1000 vehicles
D)168.199 accidents per 1000 vehicles

Accepted Answer

To find the accident rate per 1000 vehicles, divide the number of accidents (90) by the total number of vehicles (168,199) and then multiply by 1000. This calculation gives $ \frac{90}{168,199} \times 1000 \approx 0.535 $ accidents per 1000 vehicles.

Question 10

Find the expected value.
-Suppose that you arrive at a bus stop randomly, so all arrival times are equally likely. The bus arrives regularly every 10 minutes without delay. What is the expected value of your waiting time?
A)1 min
B)2 min
C)5 min
D)3 min

Accepted Answer

Since the bus arrives every 10 minutes and arrival times are uniformly distributed, the expected waiting time is half the interval between buses, which is 5 minutes.

Question 11

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-An insurance company reported that last year there were claims for medical expenses for 2629 on-the-job injuries. The company covers 329,507 employees. Find the rate of on-the-job injuries per 1000 employees. Round your answer to hundredths.
A)7.98 injuries per1000 employees
B)0.008 injury per    1000 employees.
C)0.13 injury   per 1000 employees,
D)2.629 injuries per 1000 employees.

Accepted Answer

To find the rate of on-the-job injuries per 1000 employees, divide the number of injuries (2629) by the number of employees (329,507) and then multiply by 1000. This gives $ \frac{2629}{329507} \times 1000 \approx 7.98 $ injuries per 1000 employees.

Question 12

Find the expected value.
-Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for rolling a 1 or a 2, nothing otherwise. What is your expected value of your gain or loss?
A)$0.00
B)-$1.00
C)$3.00
D)$1.00

Accepted Answer

The expected value is calculated by multiplying each outcome by its probability and then summing those values. Rolling a 1 or a 2 (2 outcomes out of 6) gives you $3.00, so the expected gain from those rolls is 2/6 * $3.00 = $1.00. For the other 4 outcomes, you gain nothing but lose your $1.00 bet, so the expected loss is 4/6 * -$1.00 = -$0.67. Adding these together, $1.00 - $0.67 = $0.33, but since you initially paid $1.00 to play, your net expected value is $0.33 - $1.00 = -$0.67. However, this calculation seems to have an error in considering the initial payment as part of the expected value calculation. Correctly, the expected gain should account for the initial bet as part of the outcomes. The correct calculation is: Expected value = (1/3 * $3.00) + (2/3 * $0) - $1.00 = $1.00 - $1.00 = $0.00.

Question 13

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-At an intersection in Normal, Illinois, there were 170 vehicle accidents with 255,872 vehicles passing through the intersection. Determine the accident rate per 10,000 vehicles.
A)6.644 accidents per 10,000 vehicles
B)255.872 accidents per 10,000 vehicles
C)1,505,129.41 accidents per 10,000 vehicles
D)0.00066 accident per 10,000 vehicles

Accepted Answer

To find the accident rate per 10,000 vehicles, divide the number of accidents (170) by the total vehicles (255,872) and then multiply by 10,000. This calculation gives $ \frac{170}{255,872} \times 10,000 \approx 6.644 $ accidents per 10,000 vehicles.

Question 14

A family with four children is randomly selected. Assume that births of boys and girls are equally likely. Construct a table showing the probability distribution for the events 4 girls, 3 girls, 2 girls, 1 girl, and 0 girls..   
$\quad $ $\quad $$\quad $$\text {4 Child Family }$
$\begin{array}{l}
\begin{array} { l l c } 
\hline { \text { Result } } && \text { Probability } \
\hline \
 \
 \
 \
 \
\hline \text { Total } & & 
\end{array}
\end{array}$

A)
  $\quad $ $\quad $$\quad $$\text {4 Child Family }$
$\begin{array}{l}
\begin{array} { l l c } 
\hline { \text { Result } } && \text { Probability } \
\hline 4 \text { girls } & 0 \text { boys } & 1 / 16 \
3 \text { girls } & 1 \text { boy } & 4 / 16 \
2 \text { girls } & 2 \text { boys } & 6 / 16 \
1 \text { girl } & 3 \text { boys } & 4 / 16 \
0 \text { girls } & 4 \text { boys } & 1 / 16 \
\hline \text { Total } & & 1
\end{array}
\end{array}$

B)  $\quad $ $\quad $$\quad $$\text {4 Child Family }$
$\begin{array}{l}
\begin{array} { l l c } 
\hline  { \text { Result } } && \text { Probability } \
\hline 4 \text { girls } & 0 \text { boys } & 1 \
3 \text { girls } & 1 \text { boy } & 1 \
2 \text { girls } & 2 \text { boys } & 1 \
1 \text { girl } & 3 \text { boys } & 1 \
0 \text { girls } & 4 \text { boys } & 1 \
\hline \text { Total } && 5
\end{array}
\end{array}$

C)
  $\quad $ $\quad $$\quad $$\text {4 Child Family }$
$\begin{array}{l}
\begin{array} { l l c } 
\hline  { \text { Result } } && \text { Probability } \
\hline 4 \text { girls } & 0 \text { boys } & 1 / 5 \
3 \text { girls } & 1 \text { boy } & 1 / 5 \
2 \text { girls } & 2 \text { boys } & 1 / 5 \
1 \text { girl } & 3 \text { boys } & 1 / 5 \
0 \text { girls } & 4 \text { boys } & 1 / 5\
\hline \text { Total } && 1
\end{array}
\end{array}$

D) 
  $\quad $ $\quad $$\quad $$\text {4 Child Family }$
$\begin{array}{l}
\begin{array} { l l c } 
\hline  { \text { Result } } && \text { Probability } \
\hline4 \text { girls } & 0 \text { boys } & 1 \
3 \text { girls } & 1 \text { boy } & 4 \
2 \text { girls } & 2 \text { boys } & 6 \
1 \text { girl } & 3 \text { boys } & 4 \
0 \text { girls } & 4 \text { boys } & 1\
\hline \text { Total } && 16
\end{array}
\end{array}$

Accepted Answer

The probabilities are calculated based on the binomial distribution formula, with each child having a 1/2 chance of being a girl or a boy. The total number of outcomes for 4 children is $2^4 = 16$. The probabilities in option A correctly reflect the number of ways each result can occur out of the 16 possible outcomes.

Question 15

Find the expected value.
-Suppose you buy 1 ticket for $1 out of a lottery of 1000 tickets where the prize for the one winning ticket is to be $10,000. What is your expected value?
A)$90.00
B)$0.90
C)$9.00
D)-$0.90

Accepted Answer

The answer of Find the expected value.
-Suppose you buy 1...

Question 16

Answer the question about the Law of Large Numbers.
-A fair coin is tossed 5000 times. What can you say about getting the outcome of exactly 2500 tails?
A. Since the probability of a tail is 0.5 for each toss, you should expect exactly 2500 tails in 5000 tosses.
B. You should not expect exactly 2500 tails in 5000 tosses, but the proportion of tails should approach 0.5 as the number
Of tosses increases.
C. You should expect between 2475 and 2525 tails in 5000 tosses.
D. Getting 2500 tails is no more likely than getting any other number of tails in 5000 tosses.
A)c
B)a
C)b
D)d

Accepted Answer

The Law of Large Numbers states that as the number of independent trials of a random experiment increases, the sample average approaches the expected value. In this case, the expected value of the number of tails is 5000 * 0.5 = 2500. Therefore, we should expect between 2475 and 2525 tails in 5000 tosses.

Question 17

Answer the question about the Law of Large Numbers.
-Jim has been stopped for speeding three times in the last year, but each time was given a warning instead of a ticket.
What does it mean to say that "the law of averages will catch up with him"?
A. It means that average drivers receive tickets, not warnings, when stopped for speeding.
B. It means that Jim is "due" to receive a ticket next time he is stopped for speeding.
C. It means that, in the long run, Jim should not expect to be stopped for minor speeding violations.
D. None of the above is a correct interpretation.
A)b
B)a
C)d
D)c

Accepted Answer

The answer of Answer the question about the Law of...

Question 18

Answer the question about the Law of Large Numbers.
-Melissa tosses a fair coin 4 times. What can we reasonably conclude about the outcome of 4 heads occurring?
A. It will occur exactly one time.
B. It will probably never occur, because it is an unusual outcome.
C. It will occur exactly 4 times.
D. None of the above is true.
A)a
B)c
C)d
D)b

Accepted Answer

DThe Law of Large Numbers suggests that the outcomes of a large number of trials will approximate the expected probability, but it does not make specific predictions about a small number of trials, such as 4 coin tosses. Therefore, none of the provided options accurately describe what we can conclude based on the Law of Large Numbers.

Question 19

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise.
-At an intersection in Normal, Illinois, there were 83 vehicle accidents with 167,231 vehicles passing through the intersection. Determine the accident rate per 100 vehicles.
A)0.00050 accident per 100 vehicles
B)0.050 accident per 100 vehicles
C)1672.31 accidents per 100 vehicles
D)201,483.13 accidents per 100 vehicles

Accepted Answer

To find the accident rate per 100 vehicles, divide the number of accidents (83) by the total number of vehicles (167,231) and then multiply by 100. This gives (83 / 167,231) * 100 = 0.0496, which rounds to 0.050 accidents per 100 vehicles.

Question 20

Find the expected value.
-A contractor is considering a sale that promises a profit of $39,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such)of $3000 with a probability of 0.3. What is the expected profit?
A)$29,400
B)$36,000
C)$27,300
D)$26,400

Accepted Answer

The answer of Find the expected value.
-A contractor is considering...

Quiz 6: Probability in Statistics

Find the expected value. -A 28-year-old man pays $57 for a one-year life insurance policy with coverage of $130,000. If the probability that he will live through the year is 0.9994, what is the expected value for the insurance policy?

Find the expected value. -In a game, you have a $\frac { 1 } { 22 }$ probability of winning $89 and a $\frac { 21 } { 22 }$ Probability of losing $9. What is your expected Value?

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise. -At an intersection in Normal, Illinois, there were 90 vehicle accidents with 168,199 vehicles passing through the intersection. Determine the accident rate per 1000 vehicles.

Find the expected value. -Suppose that you arrive at a bus stop randomly, so all arrival times are equally likely. The bus arrives regularly every 10 minutes without delay. What is the expected value of your waiting time?

Find the expected value. -Suppose you pay $1.00 to roll a fair die with the understanding that you will get back $3.00 for rolling a 1 or a 2, nothing otherwise. What is your expected value of your gain or loss?

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise. -At an intersection in Normal, Illinois, there were 170 vehicle accidents with 255,872 vehicles passing through the intersection. Determine the accident rate per 10,000 vehicles.

Find the expected value. -Suppose you buy 1 ticket for $1 out of a lottery of 1000 tickets where the prize for the one winning ticket is to be $10,000. What is your expected value?

Use the data to find the required rate. Round your answer to thousandths unless indicated otherwise. -At an intersection in Normal, Illinois, there were 83 vehicle accidents with 167,231 vehicles passing through the intersection. Determine the accident rate per 100 vehicles.

Find the expected value. -A contractor is considering a sale that promises a profit of $39,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $3000 with a probability of 0.3. What is the expected profit?