How do I find the expected value for this game?

Question

Hi, I'm having a bit of trouble trying to figure out the expected value for this game, so if you could help that would be really great and helpful. thanks so much

A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

1. If the card is a face card, and the coin lands on Heads, you win $4
2. If the card is a face card and the coin lands on Tails, you win $1
3. If the card isn't a face card, you lose $1, no matter what the coin shows

I need help finding the expected value for this game (expected net gain or loss). I also have to round the answer to two decimal places.

Answer 1

Probability of getting a face card is 12/52

Probability of coin landing on heads is 1/2

Probability of coin landing on tails is 1/2

Probability of not a face card is 40/52

Probability of face card and landing on heads is 12/52 x 1/2=12/104 or 3/26 (+$4)

Probability of face card and landing on tails is 12/52 x 1/2=12/104 or 3/26 (+$1)

Probability of not face card is 40/52 or 10/13 (-$1)

Expected value can be viewed as mean or average.

Ex=4(3/26)+1(3/26)+ -1(10/13)= -0.19

This means, that you should expect to loose, on average, 19 cents if you play the game.

Answer 2

We need to determine the probability of all occurrences and multiply them by the dollar amounts won to determine net gain or loss.

Let’s determine:

P(face card AND heads) = 12/52 * 1/2 = 0.115

P(face card AND tails) = 12/52 * 1/2 = 0.115

P(no face card) = 40/52 = 0.77

Now determine net gain or loss:

0.115 * $4 + 0.115 * $1 + 0.77 * -$1 = -$0.20 net loss