A player at a fair pays $1.50 to toss a coin three times. The player receives $1 if the number of heads is 2, $8 if the number is 3, but nothing otherwise. Find the expected value of the reward X. What is the expected value of the gain



Answer :

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.