Question 1

Find the expected value.
-Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 1, nothing otherwise. What is your expected net winnings?

Accepted Answer

The expected net winnings can be calculated by multiplying the outcomes by their probabilities and summing these products. The probability of rolling a 4 or a 1 is 2/6 (since there are 6 sides on a die and 2 favorable outcomes). The winnings for these outcomes are $2 each ($4 received minus the $2 paid to play). For the other 4 outcomes, the net loss is $2 (nothing received minus the $2 paid to play). Thus, the expected value is (2/6)*$2 + (4/6)*(-$2) = -$0.67.

Question 2

Solve the problem. Round your answer to 2 decimal places when necessary.
-A six mile strip of Sprinkle Road had 44 traffic fatalities last year. There were 2.4 million miles driven along this stretch. Determine the traffic fatality rate per 100,000 miles driven.

Accepted Answer

To find the traffic fatality rate per 100,000 miles driven, divide the number of fatalities by the total miles driven and then multiply by 100,000. This gives (44 / 2,400,000) * 100,000 = 1.83 fatalities per 100,000 miles driven.

Question 3

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-Two marbles are drawn without replacement from a box with 3 white, 2 green, 2 red, and 1 blue marble. Find the probability that both marbles are white.

Accepted Answer

The probability of drawing a white marble on the first draw is 3/8. The probability of drawing a white marble on the second draw is 2/7 (since there are now 2 white marbles and 7 total marbles left in the box). The probability of drawing both marbles white is (3/8) * (2/7) = 3/28.

Question 4

Find the indicated probability.
-The following table show the results of a clinical trial for an allergy drug. $$\begin{array} { l | c | c | c | c } 
& \begin{array} { c } 
\text { Allergy } \
\text { drug }
\end{array} & \text { Placebo } & \begin{array} { c } 
\text { Control } \
\text { (no treatment) }
\end{array} & \text { Total } \
\hline \text { Improvement } & 145 & 85 & 41 & 271 \
\text { No improvement } & 55 & 115 & 59 & 229 \
\text { Total } & 200 & 200 & 100 & 500
\end{array}$$ What is the probability that a randomly selected person was given a placebo and improved? Round your answer to the nearest thousandth when necessary.

Accepted Answer

The probability that a randomly selected person was given a placebo and improved is calculated by dividing the number of people who improved on the placebo (85) by the total number of participants (500), which gives $ \frac{85}{500} = 0.17 $.

Question 5

Use the at least once rule to find the indicated probability.
-In a batch of 8000 clock radios 3% are defective. A sample of 7 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of
Those tested is defective. What is the probability that the entire batch will be rejected?

Accepted Answer

The probability that a clock radio is defective is 0.03. The probability that a clock radio is not defective is 0.97. The probability that all 7 clock radios are not defective is 0.97^7 = 0.808. The probability that at least one clock radio is defective is 1 - 0.808 = 0.192.

Question 6

Solve the problem. Round your answer to 5 decimal places when necessary.
-The table shows the leading causes of death for one country in a single recent year. $$\begin{array} { l | r } 
 { \text { Cause of Death } } & \text { Deaths } \
\hline \text { Heart Disease } & 157,550 \
\text { Cancer } & 116,450 \
\text { AIDS } & 65,760 \
\text { Stroke } & 41,100 \
\text { Pulmonary Disease } & 38,497 \
\text { Accidents } & 35,620 \
\text { Diabetes } & 16,577 \
\text { Pneumonia } & 15,344 \
\text { Kidney Disease } & 8083
\end{array}$$ Assume a population of 68.5 million. What is the death rate due to accident in deaths per 100,000 of the population

Accepted Answer

To find the death rate due to accidents per 100,000 of the population, use the formula: $\frac{\text{Number of deaths due to accidents}}{\text{Total population}} \times 100,000$. Plugging in the numbers: $\frac{35,620}{68,500,000} \times 100,000 = 52$ deaths per 100,000.

Question 7

Use the relative frequency method to estimate the probability. Round your answer to 2 decimal places when necessary.
-You count 58 heads when you toss a coin 100 times. If you don't know whether the coin is fair, what is the probability that the next toss will be heads?

Accepted Answer

The relative frequency method estimates probability based on past outcomes. Here, with 58 heads out of 100 tosses, the estimated probability for the next toss being heads is 58/100 = 0.58.

Question 8

Find the indicated probability.
-The table shows the year and political party of a random sample of students from a certain college. $\begin{array}{r|cccr}
\text { Party } & \text { Freshman } & \text { Sophomore } & \text { Junior } & \text { Senior } \
\hline \text { Independent } & 13 & 12 & 18 & 29 \
\text { Democrat } & 20 & 29 & 12 & 13 \
\text { Republican } & 12 & 13 & 20 & 29
\end{array}$

Find the empirical probability that a student at the college is an Independent. Round your answer to the nearest thousandth.

Accepted Answer

To find the empirical probability that a student is an Independent, add the number of Independent students (13+12+18+29=72) and divide by the total number of students (13+12+18+29+20+29+12+13+12+13+20+29=216). The probability is 72/216 = 0.333, which rounds to 0.333 (option A is the closest).

Question 9

Find the expected value.
-In a large casino, the house wins on one of its games with a probability of 51%. All bets in the game are 1 : 1 . If you win, you gain the amount you bet; if you lose, you lose the amount you bet. What is
The expected value to the player of a single game?

Accepted Answer

The expected value (EV) for the player can be calculated using the formula: EV = (Probability of Winning * Amount Won per Bet) + (Probability of Losing * Amount Lost per Bet). Given the house wins with a probability of 51%, the player wins with a probability of 49%. Since the bets are 1:1, the amount won or lost per bet is equal to the bet amount, which we can consider as $1 for simplicity. Thus, EV = (0.49 * $1) + (0.51 * -$1) = $0.49 - $0.51 = -$0.02.

Question 10

Decide whether events A and B are overlapping or non-overlapping.
-A person is selected at random from a group of doctors. Event A is that the person selected is a woman. Event B is that the person selected is a surgeon.

Accepted Answer

Events A and B are overlapping because a person can be both a woman and a surgeon at the same time.

Question 11

Solve the problem. Round your answer to 2 decimal places when necessary.
-The population of a certain country is about 154 million. The overall death rate is 11.6 deaths per 1000. Approximately how many deaths are there in the country?

Accepted Answer

To find the number of deaths, multiply the population by the death rate per 1000. So, 154 million * 11.6/1000 = 1.7864 million, which rounds to approximately 1.79 million deaths.

Question 12

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-A die with 12 sides is rolled. What is the probability of rolling a number less than 11?

Accepted Answer

The answer of Find the indicated probability. Round your answer...

Question 13

Find the expected value.
-Four cards are numbered 1 through 4. Two of these cards are chosen at random without replacement and the numbers on them are multiplied. Find the expected value of this product.

Accepted Answer

The expected value can be calculated by finding the mean of all possible products. The possible pairs and their products are: (1,2) = 2, (1,3) = 3, (1,4) = 4, (2,3) = 6, (2,4) = 8, (3,4) = 12. The mean of these products is (2+3+4+6+8+12)/6 = 35/6.

Question 14

Evaluate the factorial expression.
-$\frac { 7 ! } { 9 ! }$

Accepted Answer

The factorial expression $\frac{7!}{9!}$ simplifies to $\frac{1}{8 \times 9}$ because $9! = 9 \times 8 \times 7!$. Therefore, it simplifies further to $\frac{1}{72}$.

Question 15

Solve the problem.
-There are 5 women running in a race. If a person guesses randomly the first place, second place, and third place winners, what is the Probability that they will guess all winners correctly?

Accepted Answer

The total number of ways to guess the top three winners from five runners is a permutation of 5 items taken 3 at a time, which is calculated as 5!/(5-3)!, equaling 5*4*3 = 60. Since there is only one correct way to guess the top three winners, the probability is 1 out of 60, or $\frac { 1 } { 60 }$.

Question 16

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-Find the probability of selecting either a non-smoker or a woman when you select a person at random from a group composed of 45 female smokers, 30 male smokers, 25 female non-smokers,
And 50 male non-smokers.

Accepted Answer

The answer of Find the indicated probability. Round your answer...

Question 17

Find the indicated probability. Round your answer to 6 decimal places when necessary.
-What is the probability that a family with three children does not have three children of the same gender? Assume boys and girls are equally likely.

Accepted Answer

The probability of having three children of the same gender is 1/8. Therefore, the probability of not having three children of the same gender is 1 - 1/8 = 7/8.

Question 18

Solve the problem.
-A company wants to hire a software engineer, an administrative assistant, and a sales representative. There are 4 possible candidates for the position of software engineer, 4 for the
Position of administrative assistant, and 3 for the position of sales representative. How many ways
Are there to choose the three people who will be hired?

Accepted Answer

The number of ways to choose the three people for the three positions is found by multiplying the number of candidates for each position: 4 (software engineer) * 4 (administrative assistant) * 3 (sales representative) = 48 ways.

Question 19

Solve the problem.
-In Morse code, each symbol is either a dot or a dash. How many sequences of 6 symbols are possible?

Accepted Answer

Each symbol in a sequence of 6 can be either a dot or a dash, making 2 choices for each position. Therefore, the total number of sequences is 2^6 = 64.

Question 20

Provide an appropriate response.
-Which of the following could not possibly be probabilities? A. $- 0.44$
B. $\frac { 13 } { 7 }$
C. 0
D. $0.43$

Accepted Answer

Probabilities must be between 0 and 1, inclusive. A is negative and B is greater than 1, making them impossible probabilities.

Quiz 7: Probability: Living With the Odds

Find the expected value. -Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 4 or a 1, nothing otherwise. What is your expected net winnings?

Solve the problem. Round your answer to 2 decimal places when necessary. -A six mile strip of Sprinkle Road had 44 traffic fatalities last year. There were 2.4 million miles driven along this stretch. Determine the traffic fatality rate per 100,000 miles driven.

Find the indicated probability. Round your answer to 6 decimal places when necessary. -Two marbles are drawn without replacement from a box with 3 white, 2 green, 2 red, and 1 blue marble. Find the probability that both marbles are white.

Use the relative frequency method to estimate the probability. Round your answer to 2 decimal places when necessary. -You count 58 heads when you toss a coin 100 times. If you don't know whether the coin is fair, what is the probability that the next toss will be heads?

Find the expected value. -In a large casino, the house wins on one of its games with a probability of 51%. All bets in the game are 1 : 1 . If you win, you gain the amount you bet; if you lose, you lose the amount you bet. What is The expected value to the player of a single game?

Decide whether events A and B are overlapping or non-overlapping. -A person is selected at random from a group of doctors. Event A is that the person selected is a woman. Event B is that the person selected is a surgeon.

Solve the problem. Round your answer to 2 decimal places when necessary. -The population of a certain country is about 154 million. The overall death rate is 11.6 deaths per 1000. Approximately how many deaths are there in the country?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -A die with 12 sides is rolled. What is the probability of rolling a number less than 11?

Find the expected value. -Four cards are numbered 1 through 4. Two of these cards are chosen at random without replacement and the numbers on them are multiplied. Find the expected value of this product.

Evaluate the factorial expression. - $\frac { 7 ! } { 9 ! }$

Solve the problem. -There are 5 women running in a race. If a person guesses randomly the first place, second place, and third place winners, what is the Probability that they will guess all winners correctly?

Find the indicated probability. Round your answer to 6 decimal places when necessary. -Find the probability of selecting either a non-smoker or a woman when you select a person at random from a group composed of 45 female smokers, 30 male smokers, 25 female non-smokers, And 50 male non-smokers.

Find the indicated probability. Round your answer to 6 decimal places when necessary. -What is the probability that a family with three children does not have three children of the same gender? Assume boys and girls are equally likely.

Solve the problem. -In Morse code, each symbol is either a dot or a dash. How many sequences of 6 symbols are possible?

Provide an appropriate response. -Which of the following could not possibly be probabilities? A. $- 0.44$ B. $\frac { 13 } { 7 }$ C. 0 D. $0.43$