Question

Find the expected value for this game (expected net gain or loss)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. If the card is a face card and the coin lands on heads, you win $7. If the card is a face card and the coin lands on tails, you win $3. If the card is not a face card, you lose $3, no matter what the coin shows.

A- Find the expected value for this game (expected net gain or loss).

B- Explain what your calculations indicate about your long - term average profits and losses on this game.

C- Should you play this game to win money?

*Please show all work / formulas    -Thanks

Answer 1

There are four probabilities to calculate (F=face card, N=non-face card, H=head, T=tail):

P(FH)=3/13*1/2=3/26; P(FT)=3/26; P(NH)=10/13*1/2=5/13; P(NT)=5/13.

Associated with these we have the money: +$7, +$3, -$3, -$3.

The mean expected value is 7*3/26+3*3/26-3*5/13-3*5/13=-15/13, which is an overall loss.

So, for example, in 13 games the average loss would be $15. If we look at the individual figures for 26 games, we would we get on average 3 FH and so earn $21. We would get on average 3 FT and earn $9.

But for 20 of the games we would lose in total $60. The net result is 21+9-60=-$30 a loss of $30 in 26 games, which is $15 in 13 games, on average.

The long-term result is expected to be -15n/13, a net loss, so, no, don’t play to win money!