Question

1. at a local carnival, there is a \double your money\ money booth where players grab fake $5 bills floating in the air. the objective is to grab as many bills as possible in ten seconds. after the game is played, each fake $5 bill is exchanged for $10 in real cash. the game costs $50 to play. let f represent the total fake dollar amount grabbed in one play of the game. the probability distribution of the value of f is shown in the table.

f = value of fake bills ($)	$5	$10	$15	$20	$25	$30	$35

the mean of the distribution of f is $23.50 with a standard deviation of $7.16. after playing the game, the player then converts the fake bills into real dollars and collects their overall winnings (w) using the following conversion w = 2f - 50.

a. what is the mean of w?

b. what is the standard deviation of w?

c. another game at the carnival is \duck pond\, where players choose a rubber duck floating in a small pool and win the dollar amount written on the bottom of the duck. let d represent the winnings, in dollars, from one play of the \duck pond\ game. the mean of d is -$2.25 and the standard deviation is $4.22. suppose a person plays the double your money game once and the duck pond game once. what is the mean amount of the total winnings from playing these two games?

d. what is the standard deviation of the amount of the total winnings from playing these two games?

math medic

Explanation:

Part (a)

Step 1: Recall the formula for the mean of a linear transformation

If \( W = aF + b \), then the mean of \( W \), denoted \( \mu_W \), is given by \( \mu_W = a\mu_F + b \). Here, \( W = 2F - 50 \), so \( a = 2 \), \( b = -50 \), and \( \mu_F = 23.50 \).

Step 2: Substitute the values into the formula

Substitute \( a = 2 \), \( \mu_F = 23.50 \), and \( b = -50 \) into the formula: \( \mu_W = 2(23.50) - 50 \).
First, calculate \( 2(23.50) = 47 \). Then, subtract 50: \( 47 - 50 = -3 \).

Step 1: Recall the formula for the standard deviation of a linear transformation

If \( W = aF + b \), the standard deviation of \( W \), denoted \( \sigma_W \), is given by \( \sigma_W = |a|\sigma_F \). Here, \( a = 2 \) and \( \sigma_F = 7.16 \).

Step 2: Substitute the values into the formula

Substitute \( a = 2 \) and \( \sigma_F = 7.16 \) into the formula: \( \sigma_W = |2| \times 7.16 \).
Since \( |2| = 2 \), we have \( 2 \times 7.16 = 14.32 \).

Step 1: Define the total winnings

Let \( T \) be the total winnings from playing both games, so \( T = W + D \), where \( W \) is the winnings from "Double Your Money" and \( D \) is the winnings from "Duck Pond".

Step 2: Recall the formula for the mean of a sum of random variables

The mean of the sum of two random variables \( X \) and \( Y \) is \( \mu_{X + Y} = \mu_X + \mu_Y \). Here, \( X = W \) and \( Y = D \). We know \( \mu_W = -3 \) (from part (a)) and \( \mu_D = -2.25 \).

Step 3: Substitute the values into the formula

Substitute \( \mu_W = -3 \) and \( \mu_D = -2.25 \) into the formula: \( \mu_T = \mu_W + \mu_D = -3 + (-2.25) \).
Calculate the sum: \( -3 - 2.25 = -5.25 \).

Answer:

The mean of \( W \) is \(\$ -3\) (or \(-\$3\)).

Part (b)