Step-by-step explanation:
let's see the possible events when tossing a coin 3 times (h = heads, t = tails) :
t t t
t t h
t h t
h t t
t h h
h t h
h h t
h h h
as expected, we have 2³ = 8 different events, as each toss has 2 possible single events, abd then we do this 3 independent times : 2×2×2
so, now, each event has 0 value unless it has 2 or 3 heads.
that means the list of value is (result minus the playing fee) :
0 - 1.50 = -1.50
0 - 1.50 = -1.50
0 - 1.50 = -1.50
0 - 1.50 = -1.50
1 - 1.50 = -0.50
1 - 1.50 = -0.50
1 - 1.50 = -0.50
8 - 1.50 = 6.50
the expected value is simply the mean value (sum of all data points divided by the number of data points) :
(4×-1.5 + 3×-0.5 + 6.5)/8 = -1/8 = -0.125
that simply means : overall, the bank will always win, even as in 1/8 of the games it has to pay big.
