Question

4 numeric 10 points
a dressmaker just created dresses to be sold. there is a \\(\frac{3}{11}\\) chance that they will be sold today, in which case the profit will be $72. there is a \LXI1 chance that they will not be sold, in which case the dressmaker will lose $33. find the dressmaker’s expected profit or loss. round to the nearest penny.
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5 numeric 10 points
how many ways can a club select a president, vice president, secretary, and treasurer from 10 members?
answer

6 numeric 10 points
james is in a gambling casino that charges $5 per chance to roll an honest die. he will be paid, in dollars, the number of dots on the face of the die. find his expected profit or loss. round to the nearest penny.
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Explanation:

Question 4

Step1: Recall expected value formula

The expected value \( E(X) \) is calculated as \( E(X)=\sum x_iP(x_i) \), where \( x_i \) are the outcomes and \( P(x_i) \) are their probabilities. Here, the two outcomes are profit of \( \$72 \) (probability \( \frac{3}{11} \)) and loss of \( \$33 \) (which is \( -\$33 \), probability \( \frac{8}{11} \)).

Step2: Calculate expected value

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Step1: Identify the problem type

This is a permutation problem because the order of selection (president, vice - president, secretary, treasurer) matters. The formula for permutations is \( P(n,r)=\frac{n!}{(n - r)!} \), where \( n = 10 \) (total number of members) and \( r=4 \) (number of positions to fill).

Step2: Calculate the permutation

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Step1: Recall expected value formula

The expected value of the die roll \( E(X) \) (the amount he is paid) and then we subtract the cost of \( \$5 \) to find the expected profit/loss. The die has outcomes \( 1,2,3,4,5,6 \), each with probability \( \frac{1}{6} \).

Step2: Calculate expected value of die roll

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Step3: Calculate expected profit/loss

The cost is \( \$5 \), so the expected profit/loss \( E = E(\text{die})-5=3.5 - 5=-1.50 \)

Answer:

\(-\$4.36\) (or a loss of \$4.36)

Question 5