Question

you pick a card from a standard deck. if you get a face card (jack, queen or king), you win $10. if you get an ace, you win $20 plus an extra $40 for the ace of hearts. for any other card you win nothing. find the expected value of the amount you will win.

Explanation:

Step1: Calculate the probability of each event

• A standard deck has 52 cards.
• There are 12 face cards (4 Jacks, 4 Queens, 4 Kings), so \( P(\text{face card})=\frac{12}{52}=\frac{3}{13} \).
• There are 4 aces, so \( P(\text{ace})=\frac{4}{52}=\frac{1}{13} \).
• There is 1 ace of hearts, so \( P(\text{ace of hearts})=\frac{1}{52} \).
• For other cards (non - face, non - ace), there are \( 52-(12 + 4)=36 \) cards, so \( P(\text{other card})=\frac{36}{52}=\frac{9}{13} \).

Step2: Calculate the expected value

The formula for expected value \( E(X)=\sum_{i}x_{i}P(x_{i}) \)

• If it's a face card (\( x_1 = 10 \)), \( P(x_1)=\frac{3}{13} \).
• If it's an ace (excluding ace of hearts, \( x_2 = 20 \)), \( P(x_2)=\frac{4 - 1}{52}=\frac{3}{52} \).
• If it's the ace of hearts (\( x_3=20 + 40=60 \)), \( P(x_3)=\frac{1}{52} \).
• If it's another card (\( x_4 = 0 \)), \( P(x_4)=\frac{9}{13} \).

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Answer:

The expected value is approximately \(\$4.62\)