What is your expected value?

Question

In a game you bet $1 and toss two coins. If two heads come up, you win a profit of 75 cents. If two tails come up, no one wins anything. Otherwise, you lose your dollar. What is your expected value for this game? Don’t round your answer.

Answer 1

Adrienne,

Such problems are intended to teach you concepts and formulas that you can use to quickly solve probability problems in the future.

You couldtoss two coins many, many times and record the experimental results, or you can get the expected value if you assume:

  1.   the coins are independent  (that is, the result of one coin does not influence the value of the other)
  2.  each coin is  “honest.”  That is, the probabilities of coming up heads or tails are equal)

Note that for each value, heads or tails, of the first coin, the second coin can independently have the value of heads or tails.  That means there are 2*2 = 4 possible results (equally likely).

Now, of the 4 possible outcomes, here is how we do:
          H  H    keep $1 and win profit of  75 cents
          H  T   lose $1
          T  H   lose $1
          T  T    no one wins anything

So, if you play the game N times, you can expect (in cents):
        E = (1/4)*175 + (1/4)*(-100) + (1/4)*(-100) + (1/4) (0)  
        E = (1/4)*(-25) = -6.25
       For N tries at this game, I expect to lose 6.25*N cents.