Question

a player pays $15 to play a game in which a chip is randomly selected from a bag of chips. the bag contains 10 red chips, 4 blue chips, and 6 yellow chips. the player wins $5 if a red chip is selected, $10 if a blue chip is selected, and $20 if a yellow chip is selected. let the random variable ( x ) represent the amount won from the selection of the chip, and let the random variable ( w ) represent the total amount won, where ( w = x - 15 ). what is the mean of ( w )?

Explanation:

Step1: Calculate total number of chips

Total chips = $10 + 4 + 6 = 20$

Step2: Find probabilities of each outcome

$P(\text{red}) = \frac{10}{20} = 0.5$, $P(\text{blue}) = \frac{4}{20} = 0.2$, $P(\text{yellow}) = \frac{6}{20} = 0.3$

Step3: Compute mean of $X$

$\mu_X = (5 \times 0.5) + (10 \times 0.2) + (20 \times 0.3)$
$\mu_X = 2.5 + 2 + 6 = 10.5$

Step4: Apply linearity of expectation to $\mu_W$

Since $W = X - 15$, $\mu_W = \mu_X - 15$
$\mu_W = 10.5 - 15 = -4.5$

Answer:

$-4.5$