Question

If you draw a card with a value of five or less from a standard deck of cards, I will pay you $39. If not, you pay me $22. (Aces are considered the highest card in the deck.) Step 1 of 2: Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Answer 1

A deck has 52 cards with 4 suits. Each suit has 13 cards from 2 - 10 and J, Q, K and Ace

Therefore number of cards less than or equal to the value of 5 are 2, 3, 4 and 5 and there are 4 of each so 16 cards. Probability of drawing these 16 cards = 16/52 = 4/13

The remaining 52 - 16 = 36 cards are greater than 5. The probability of drawing a card from these = 36/52 = 9/13.

You win $39 if you pick a card with probability 4/13 and you lose -$22 (minus indicates loss) if you pick a card with probability 9/13

Therefore Expected value = Sum [x * P(x)] = [39 * (4/13)] - [22 * (9/13)] = (12 - 15.23) = -3.23