expectation is defined as #sum_i^n P(x)x#. In this case we know that the possible outcomes for a round is
- x_1 =3 heads, $15
- x_2 = 3 tails, $10
- x_3 =anything else, 0$
so #P(x=x_1)=1/8# because there is only 1 way to accomplish this.
#P(x=x_2)=1/8# because this is the only way for this one too.
this means that #P(x=x_3)=6/8#
for two games we could have
#P(x=x_1, x= x_1)=1/64, 30#
#P(x=x_1, x= x_2)=1/64, 25#
#P(x=x_1, x= x_3)=6/64, 15#
#P(x=x_2, x= x_1)=1/64, 25#
#P(x=x_2, x= x_2)=1/64, 20#
#P(x=x_2, x= x_3)=6/64, 10#
#P(x=x_3, x= x_1)=6/64, 15#
#P(x=x_3, x= x_2)=6/64, 10#
#P(x=x_3, x= x_3)=36/64, 0#
Leading to
#1/64*30 + 1/64*25 + 6/64*15 + 1/64* 25 +
1/64*20 + 6/64* 10 + 6/64* 25 + 6/64* 20#
simplifying
#1/64(30 + 25 + 25 +20) + 6/64(15 + 10 + 15 + 10)#
#1/64(100) + 6/64(50) = 400/64 = 6.25#
Another way to think about this is knowing that the expectation for a two games can be stated as #E(game1+game2) = E(game1) +E(game2)# thus #15*1/8 + 10 *1/8 +0*6/8 = 25/8# for one game.
for two games it should be #50/8 = 6.25# which it is. Now we can subtract the amount to play per game leaving
#6.25 - 10 = 3.75#. Playing the two games should result in a loss of #$3.75#