Question

A standard deck of 52 cards contains 13 hearts. Consider this wager: draw one card at random from the deck. If the card drawn is a heart, you win $2. Otherwise, you lose $1. What is the expected value of this wager?

Answer 1

The expected value of the wager where you win $2 for drawing a heart and lose $1 for any other card from a standard deck of 52 cards is -$0.25. This means, on average, you'd lose 25 cents per game over time.

Step-by-step explanation:

You've asked about the expected value of a wager involving drawing a card from a standard deck of 52 cards. To calculate this, we assess the two possible outcomes: drawing a heart and winning $2, or drawing any other card and losing $1.

Here's the step-by-step explanation:

1. First, calculate the probability of drawing a heart, which is 13 hearts in a 52-card deck. So, P(heart) = 13/52 = 1/4.
2. Next, calculate the probability of not drawing a heart, which is P(not a heart) = 1 - P(heart) = 1 - 1/4 = 3/4.
3. Then, determine the expected value (EV) using the formula: EV = (winning amount x probability of winning) + (losing amount x probability of losing).
4. Plugging in the values: EV = ($2 x 1/4) + (-$1 x 3/4) = $0.50 - $0.75 = -$0.25.

The expected value of the wager is -$0.25, which means you would expect to lose, on average, 25 cents per game if you played this wager repeatedly over time.