Question

you are challenged to a lucky draw game. if you draw a face card (k, q, j) from a standard deck of cards, you earn 10 points. if you draw any other card, you lose 2 points. what is the expected value of a draw?
○ 0.77
○ 1.69
○ 1.85
○ partially visible option

Explanation:

Step1: Determine number of face cards and total cards

A standard deck has 52 cards. Face cards (K, Q, J) have 3 ranks, 4 suits each: \( 3 \times 4 = 12 \) face cards. Non - face cards: \( 52 - 12 = 40 \) cards.

Step2: Calculate probabilities

Probability of face card (\( P(\text{face}) \)): \( \frac{12}{52}=\frac{3}{13} \). Probability of non - face card (\( P(\text{non - face}) \)): \( \frac{40}{52}=\frac{10}{13} \).

Step3: Apply expected value formula

Expected value \( E = (10)\times P(\text{face})+(- 2)\times P(\text{non - face}) \).
Substitute values: \( E = 10\times\frac{3}{13}+(-2)\times\frac{10}{13}=\frac{30}{13}-\frac{20}{13}=\frac{10}{13}\approx0.77 \).

Answer:

0.77