Question

3 0 / 14 points fill in the blank a dressmaker just created dresses to be sold. there is a \\( \frac{5}{11} \\) chance that they will be sold today, in which case the profit will be \\$76. there is a \LXI1 chance that they will not be sold, in which case the dressmaker will lose \\$31. find the dressmaker’s expected value. round to the nearest penny. \\( \\$ \\) \\( \boldsymbol{\times} \\) 27.36

Explanation:

Step1: Recall Expected Value Formula

The expected value \( E(X) \) is calculated as \( E(X)=\sum_{i} x_i P(x_i) \), where \( x_i \) are the possible outcomes and \( P(x_i) \) are their corresponding probabilities.
Here, the two outcomes are:

• Selling: \( x_1 = 76 \), \( P(x_1)=\frac{5}{11} \)
• Not selling: \( x_2=- 31 \) (negative because it's a loss), \( P(x_2)=\frac{6}{11} \)

Step2: Calculate Each Term

First term (selling): \( x_1P(x_1)=76\times\frac{5}{11}=\frac{380}{11}\approx34.5455 \)
Second term (not selling): \( x_2P(x_2)=- 31\times\frac{6}{11}=\frac{- 186}{11}\approx - 16.9091 \)

Step3: Sum the Terms

\( E(X)=\frac{380}{11}+\frac{-186}{11}=\frac{380 - 186}{11}=\frac{194}{11}\approx17.64 \)

Answer:

\( 17.64 \)