Question

amy has just opened a new store. if successful, she can expect an annual income of $75,000. if unsuccessful, she will lose $95,000. if the probability of success is 0.85, find her mathematical expectation for the first year. round to the nearest dollar.
$ type your answer...
2 fill in the blank 14 points
a card is drawn at random from an ordinary deck of cards. if the card selected is a face card, then amy wins $15. if the card selected is an ace, then amy wins $20. otherwise, she loses $10. what is her expected value? round to the nearest penny
$ type your answer...
3 fill in the blank 14 points
a dressmaker just created dresses to be sold. there is a \\(\frac{6}{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.
$ type your answer...

Explanation:

Problem 1:

Step1: Identify probabilities and values

Success probability \( p_1 = 0.85 \), success income \( v_1 = 75000 \). Failure probability \( p_2 = 1 - 0.85 = 0.15 \), failure loss \( v_2 = -95000 \) (negative for loss).

Step2: Calculate expectation

Expectation \( E = p_1v_1 + p_2v_2 \)
\( E = 0.85 \times 75000 + 0.15 \times (-95000) \)
\( E = 63750 - 14250 = 49500 \)

Step1: Determine probabilities

Deck has 52 cards. Face cards: 12 (4J,4Q,4K), aces: 4, others: \( 52 - 12 - 4 = 36 \).
\( p_{\text{face}} = \frac{12}{52} \), \( p_{\text{ace}} = \frac{4}{52} \), \( p_{\text{other}} = \frac{36}{52} \).
Values: face win \( 15 \), ace win \( 20 \), other lose \( -10 \).

Step2: Compute expectation

\( E = \frac{12}{52} \times 15 + \frac{4}{52} \times 20 + \frac{36}{52} \times (-10) \)
Simplify: \( E = \frac{180 + 80 - 360}{52} = \frac{-100}{52} \approx -1.923 \), round to nearest penny: \( -1.92 \)

Step1: Identify probabilities and values

Sell probability \( p_1 = \frac{6}{11} \), sell profit \( v_1 = 76 \). Not sell probability \( p_2 = \frac{5}{11} \), not sell loss \( v_2 = -31 \) (negative for loss).

Step2: Calculate expectation

\( E = \frac{6}{11} \times 76 + \frac{5}{11} \times (-31) \)
\( E = \frac{456 - 155}{11} = \frac{301}{11} \approx 27.3636 \), round to nearest penny: \( 27.36 \)

Answer:

\( \$49500 \)

Problem 2: