Question 1

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).Let X be the sum of the two cards.Find the expected value of X.

Accepted Answer

The answer of Consider a game that consists of dealing...

Question 2

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the standard deviation of the number of children the couple have.Assume that boys and girls are equally likely.Round your answer to three decimal places.

Accepted Answer

Let X be the number of children the couple have until they get a boy. Then X follows a geometric distribution with p = 1/2 (assuming boys and girls are equally likely). The mean of X is given by E(X) = 1/p = 2, and the variance is given by Var(X) = (1-p)/p^2 = 2.

However, we also have the condition that the couple will not have more than four children, even if all are girls. In other words, X is truncated at 4. We can adjust for this by using the truncated geometric distribution. Using the formula for the standard deviation of a truncated geometric distribution, we get:

SD(X) = sqrt{ Var(X) - [1-F(4)]/p^2 } = sqrt{2 - [1-1/16]/(1/4)^2 } = 1.053 (rounded to three decimal places)

Therefore, the answer is C, 1.053.

Question 3

A company bids on two contracts.It anticipates a profit of $70,000 if it gets the larger contract and a profit of $30,000 if it gets the smaller contract.It estimates that there's a 10% chance of winning the larger contract and a 60% chance of winning the smaller contract.Find the standard deviation of the company's profit.Assume that the contracts will be awarded independently.

Accepted Answer

The answer of A company bids on two contracts.It anticipates...

Question 4

The random variable x is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate Office.Its probability distribution is as follows.Find the standard deviation of the number of houses sold. $$\begin{array} { c | c } 
\text { Houses Sold } ( \mathrm { x } ) & \text { Probability P(x) } \
\hline 0 & 0.24 \
\hline 1 & 0.01 \
\hline 2 & 0.12 \
\hline 3 & 0.16 \
\hline 4 & 0.01 \
\hline 5 & 0.14 \
\hline 6 & 0.11 \
\hline 7 & 0.21
\end{array}$$

Accepted Answer

To find the standard deviation of the number of houses sold, we first need to calculate the mean ($\mu$) and then use it to find the variance ($\sigma^2$). The standard deviation ($\sigma$) is the square root of the variance.1. Calculate the mean ($\mu$):$\mu = \sum [x \cdot P(x)] = (0 \cdot 0.24) + (1 \cdot 0.01) + (2 \cdot 0.12) + (3 \cdot 0.16) + (4 \cdot 0.01) + (5 \cdot 0.14) + (6 \cdot 0.11) + (7 \cdot 0.21) = 3.59$2. Calculate the variance ($\sigma^2$):$\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$$\sigma^2 = (0-3.59)^2 \cdot 0.24 + (1-3.59)^2 \cdot 0.01 + (2-3.59)^2 \cdot 0.12 + (3-3.59)^2 \cdot 0.16 + (4-3.59)^2 \cdot 0.01 + (5-3.59)^2 \cdot 0.14 + (6-3.59)^2 \cdot 0.11 + (7-3.59)^2 \cdot 0.21$$\sigma^2 = 6.8641$3. Calculate the standard deviation ($\sigma$):$\sigma = \sqrt{\sigma^2} = \sqrt{6.8641} = 2.62$Therefore, the standard deviation of the number of houses sold is 2.62.

Question 5

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card).If you get a club the second time,you win $30.If not,you lose.Find the standard deviation of the amount you will win.

Accepted Answer

The standard deviation is calculated using the probabilities and outcomes: P(Win $80) = 1/4, P(Win $30) = 3/4 * 1/4, P(Lose $0) = 3/4 * 3/4. The expected value is $27.5. The variance is calculated as the sum of squared differences from the expected value, weighted by probabilities, and the standard deviation is the square root of the variance, which is approximately $33.35.

Question 6

$$\begin{array} { c | r c r } 
\mathrm { x } & 0 & 1 & 2 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.7 & 0.2 & 0.1
\end{array}$$

Accepted Answer

P(X = 1 or X = 2) = P(X = 1) + P(X = 2) = 0.2 + 0.1 = 0.3

Question 7

A carnival game offers a $80 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play and you're willing to spend up to $20 trying to win.You estimate that you have a 8% chance of hitting the balloon on any throw.Find the standard deviation of the number of darts you throw.Assume that throws are independent of each other.

Accepted Answer

The standard deviation is the square root of the variance. The variance is the expected value of the squared deviation from the mean. The mean number of darts thrown is 20/0.08 = 250. The variance is 250*0.08*0.92 = 184. The standard deviation is the square root of 184, which is 13.56.

Question 8

A teacher grading statistics homework finds that none of the students has made more than three errors.14% have made three errors,25% have made two errors,and 39% have made one error.Find the standard deviation of the number of errors in students' statistics homework.

Accepted Answer

The standard deviation is the square root of the variance. The variance is the average of the squared deviations from the mean. The mean is the average number of errors. The variance is calculated as follows:Variance = (0.14 * (3 - 1.4)^2) + (0.25 * (2 - 1.4)^2) + (0.39 * (1 - 1.4)^2) + (0.22 * (0 - 1.4)^2) = 0.94The standard deviation is the square root of the variance, which is 0.97.

Question 9

The probabilities that a batch of 4 computers will contain 0,1,2,3,and 4 defective computers are 0.4746,0.3888,0.1195,0.0163,and 0.0008,respectively.Find the standard deviation of the number of defective computers.

Accepted Answer

The standard deviation is the square root of the variance. The variance is the expected value of the squared deviation from the mean. The mean is the expected value. The expected value is the sum of the products of the values and their probabilities. The variance is the sum of the products of the squared deviations from the mean and their probabilities. The standard deviation is the square root of the variance.The mean is 0.4746*0 + 0.3888*1 + 0.1195*2 + 0.0163*3 + 0.0008*4 = 0.9492.The variance is 0.4746*(0 - 0.9492)^2 + 0.3888*(1 - 0.9492)^2 + 0.1195*(2 - 0.9492)^2 + 0.0163*(3 - 0.9492)^2 + 0.0008*(4 - 0.9492)^2 = 0.5625.The standard deviation is the square root of 0.5625, which is 0.75.

Question 10

A police department reports that the probabilities that 0,1,2,and 3 burglaries will be reported in a given day are 0.52,0.42,0.05,and 0.01,respectively.Find the standard deviation of the number of burglaries in a day.

Accepted Answer

The answer of A police department reports that the probabilities...

Question 11

You roll a pair of dice.If you get a sum greater than 10 you win $50.If you get a double you win $20.If you get a double and a sum greater than 10 you win a $70.Otherwise you win nothing.Find the standard deviation of the amount you win at this game.

Accepted Answer

The possible outcomes and their probabilities are calculated, then the expected value and variance are determined. The standard deviation is the square root of the variance, which is approximately $16.56.

Question 12

$$\begin{array} { l | c c c c } 
\mathrm { x } & 100 & 200 & 300 & 400 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.4 & 0.3 & 0.1
\end{array}$$

Accepted Answer

The answer of \[\begin{array} { l | c c c...

Question 13

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. $$\begin{array} { l | c c c c } 
\text { Number of storms } & 0 & 1 & 2 & 3 \
\hline \text { Probability } & 0.1 & 0.2 & 0.4 & 0.3
\end{array}$$ What is the standard deviation of the number of storms in August?

Accepted Answer

To find the standard deviation, first calculate the mean (expected value) of the number of storms: $E(X) = 0(0.1) + 1(0.2) + 2(0.4) + 3(0.3) = 1.7$. Then, calculate the variance: $Var(X) = (0-1.7)^2(0.1) + (1-1.7)^2(0.2) + (2-1.7)^2(0.4) + (3-1.7)^2(0.3) = 0.81$. The standard deviation is the square root of the variance: $\sqrt{0.81} = 0.9$. The closest answer is 0.94.

Question 14

Find the standard deviation for the given probability distribution. $$\begin{array} { c | c } 
\mathrm { x } & \mathrm { P } ( \mathrm { x } ) \
\hline 0 & 0.27 \
\hline 1 & 0.07 \
\hline 2 & 0.07 \
\hline 3 & 0.22 \
\hline 4 & 0.37
\end{array}$$

Accepted Answer

The answer of Find the standard deviation for the given...

Question 15

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $20 plus an extra $40 for the ace of hearts.For any other card you win nothing.Find the standard deviation of the amount you will win.

Accepted Answer

The standard deviation is $10.53. This is because the probability of getting a face card is 3/13, the probability of getting an ace is 1/13, and the probability of getting any other card is 9/13. The expected value of the amount you will win is $15 * 3/13 + $20 * 1/13 + $40 * 1/52 = $10.53. The variance of the amount you will win is $15^2 * 3/13 + $20^2 * 1/13 + $40^2 * 1/52 - $10.53^2 = $111.23. The standard deviation is the square root of the variance, which is $10.53.

Question 16

Jo is a hairstylist.The probability model below describes the number of clients that she may see in a day. $$\begin{array} { l | c c c c c c } 
\text { Number of clients } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \text { Probability } & 0.05 & 0.15 & 0.15 & 0.3 & 0.2 & 0.15
\end{array}$$ What is the standard deviation of the number of clients that Jo sees per day?

Accepted Answer

The answer of Jo is a hairstylist.The probability model below...

Question 17

Sue Anne owns a medium-sized business.The probability model below describes the number of employees that may call in sick on any given day. $$\begin{array} { c | c c c c c } 
\text { Number of Employees Sick } & 0 & 1 & 2 & 3 & 4 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.05 & 0.4 & 0.25 & 0.2 & 0.1
\end{array}$$ What is the standard deviation of the number of employees calling in sick each day?

Accepted Answer

The answer of Sue Anne owns a medium-sized business.The probability...

Question 18

$$\begin{array} { c | c c c r } 
\mathrm { x } & 3 & 6 & 9 & 12 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3
\end{array}$$

Accepted Answer

The answer of \[\begin{array} { c | c c c...

Question 19

The accompanying table describes the probability distribution for the number of adults in a certain town (among 4 randomly selected adults)who have a degree from a post-secondary institute.Find the standard deviation for the probability distribution. $$\begin{array} { c | c } 
\mathrm { x } & \mathrm { P } ( \mathrm { x } ) \
\hline 0 & 0.0256 \
\hline 1 & 0.1536 \
\hline 2 & 0.3456 \
\hline 3 & 0.3456 \
\hline 4 & 0.1296
\end{array}$$

Accepted Answer

To find the standard deviation of a probability distribution, we first need to calculate the mean (μ) and then use it to calculate the variance (σ^2). The standard deviation (σ) is the square root of the variance.1. Calculate the mean (μ):$$μ = \sum (x \cdot P(x)) = (0 \cdot 0.0256) + (1 \cdot 0.1536) + (2 \cdot 0.3456) + (3 \cdot 0.3456) + (4 \cdot 0.1296) = 2.4$$2. Calculate the variance (σ^2):$$σ^2 = \sum [(x - μ)^2 \cdot P(x)]$$$$= [(0-2.4)^2 \cdot 0.0256] + [(1-2.4)^2 \cdot 0.1536] + [(2-2.4)^2 \cdot 0.3456] + [(3-2.4)^2 \cdot 0.3456] + [(4-2.4)^2 \cdot 0.1296]$$$$= 1.3824$$3. Calculate the standard deviation (σ):$$σ = \sqrt{σ^2} = \sqrt{1.3824} ≈ 0.98$$Therefore, the standard deviation of the probability distribution is approximately 0.98.

Question 20

$$\begin{array} { c | c c c } 
\mathrm { x } & 20 & 40 & 60 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.3 & 0.5
\end{array}$$

Accepted Answer

Without any context or information about what the alphanumeric string represents, it is impossible to determine the correct answer. The provided characters do not contain any numerical values or units of measurement. Therefore, the answer cannot be calculated or inferred.

Quiz 14: Random Variables

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the standard deviation of the number of children the couple have.Assume that boys and girls are equally likely.Round your answer to three decimal places.

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card) .If you get a club the second time,you win $30.If not,you lose.Find the standard deviation of the amount you will win.

$\begin{array} { c | r c r } \mathrm { x } & 0 & 1 & 2 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.7 & 0.2 & 0.1\end{array}$

A teacher grading statistics homework finds that none of the students has made more than three errors.14% have made three errors,25% have made two errors,and 39% have made one error.Find the standard deviation of the number of errors in students' statistics homework.

The probabilities that a batch of 4 computers will contain 0,1,2,3,and 4 defective computers are 0.4746,0.3888,0.1195,0.0163,and 0.0008,respectively.Find the standard deviation of the number of defective computers.

A police department reports that the probabilities that 0,1,2,and 3 burglaries will be reported in a given day are 0.52,0.42,0.05,and 0.01,respectively.Find the standard deviation of the number of burglaries in a day.

You roll a pair of dice.If you get a sum greater than 10 you win $50.If you get a double you win $20.If you get a double and a sum greater than 10 you win a $70.Otherwise you win nothing.Find the standard deviation of the amount you win at this game.

$\begin{array} { l | c c c c } \mathrm { x } & 100 & 200 & 300 & 400 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.4 & 0.3 & 0.1\end{array}$

Find the standard deviation for the given probability distribution. $\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.27 \\\hline 1 & 0.07 \\\hline 2 & 0.07 \\\hline 3 & 0.22 \\\hline 4 & 0.37\end{array}$

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $20 plus an extra $40 for the ace of hearts.For any other card you win nothing.Find the standard deviation of the amount you will win.

$\begin{array} { c | c c c r } \mathrm { x } & 3 & 6 & 9 & 12 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3\end{array}$

$\begin{array} { c | c c c } \mathrm { x } & 20 & 40 & 60 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.3 & 0.5\end{array}$

Quiz 14: Random Variables

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the standard deviation of the number of children the couple have.Assume that boys and girls are equally likely.Round your answer to three decimal places.

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card) .If you get a club the second time,you win $30.If not,you lose.Find the standard deviation of the amount you will win.

x012P(X=x) 0.70.20.1\begin{array} { c | r c r } \mathrm { x } & 0 & 1 & 2 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.7 & 0.2 & 0.1\end{array}xP(X=x) ​00.7​10.2​20.1​​

A teacher grading statistics homework finds that none of the students has made more than three errors.14% have made three errors,25% have made two errors,and 39% have made one error.Find the standard deviation of the number of errors in students' statistics homework.

The probabilities that a batch of 4 computers will contain 0,1,2,3,and 4 defective computers are 0.4746,0.3888,0.1195,0.0163,and 0.0008,respectively.Find the standard deviation of the number of defective computers.

A police department reports that the probabilities that 0,1,2,and 3 burglaries will be reported in a given day are 0.52,0.42,0.05,and 0.01,respectively.Find the standard deviation of the number of burglaries in a day.

You roll a pair of dice.If you get a sum greater than 10 you win $50.If you get a double you win $20.If you get a double and a sum greater than 10 you win a $70.Otherwise you win nothing.Find the standard deviation of the amount you win at this game.

x100200300400P(X=x) 0.20.40.30.1\begin{array} { l | c c c c } \mathrm { x } & 100 & 200 & 300 & 400 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.4 & 0.3 & 0.1\end{array}xP(X=x) ​1000.2​2000.4​3000.3​4000.1​​

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $20 plus an extra $40 for the ace of hearts.For any other card you win nothing.Find the standard deviation of the amount you will win.

x36912P(X=x) 0.10.40.20.3\begin{array} { c | c c c r } \mathrm { x } & 3 & 6 & 9 & 12 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3\end{array}xP(X=x) ​30.1​60.4​90.2​120.3​​

x204060P(X=x) 0.20.30.5\begin{array} { c | c c c } \mathrm { x } & 20 & 40 & 60 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.3 & 0.5\end{array}xP(X=x) ​200.2​400.3​600.5​​

$\begin{array} { c | r c r } \mathrm { x } & 0 & 1 & 2 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.7 & 0.2 & 0.1\end{array}$

$\begin{array} { l | c c c c } \mathrm { x } & 100 & 200 & 300 & 400 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.4 & 0.3 & 0.1\end{array}$

$\begin{array} { c | c c c r } \mathrm { x } & 3 & 6 & 9 & 12 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3\end{array}$

$\begin{array} { c | c c c } \mathrm { x } & 20 & 40 & 60 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.3 & 0.5\end{array}$