Suppose you flip a coin 3 times, and record the sequence of heads and tails that appear.
You win $3, if three heads appear, $2 if two heads appear, and $1 if one head appears, but lose $10 if no heads appear. Is this game fair? why or why not, and what is the expected value?
The three coins could land any these 8 ways:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
P(3 heads) = 1 way out of 8 or 1/8
P(2 heads) = 3 ways out of 8 or 3/8
P(1 head) = 3 ways out of 8 or 3/8
P(0 heads) = 1 way out of 8 or 1/8
x=Winnings P(x) E(x)=x·P(x)
$3 1/8 $.375
$2 3/8 $.75
$1 3/8 $.375
-$10 1/8 -$1.25
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Total expectation = $ .25
The game is not fair because the total
expectation is $.25 and not 0. [If you had
to pay a quarter to play the game, then it
would be fair.]
Edwin