Question

If you throw exactly one head in three tosses of a coin you win $87. If not, you pay me $48. Enter the expected value of the proposition. Round your answer to two decimal places.

Answer 1

The number X of heads in three tosses of a coin has binomial distribution with

p=0.5, q=1-p=0.5 and n=3.


Then the expected value of the
proposition is

E = 87 * P(X=1) + 48 *P(X<>1).

We have


P(X=1) = C_3^1 p q^2,

where
C_3^1 = 3!/(1! 2!) = 6 / 2 =
3
is the binomial coefficient.

Hence

P(X=1) = 3 * 0.5 * 0.5^2
= 3/8

Therefore

P(X<>1) = 1-P(X=1) = 1-3/8 =
5/8.


Therefore

E = 87 * P(X=1) + 48 *P(X<>1)
=
= 87 * 3/8 + 48 * 5/8 =
= 62.625
~ 62.63.