Question

5 coins are tossed simultaneously. it costs nothing to play. if 1,2,3,4 heads occur, you will be paid 1 dollar for each head. if all heads or all tails occurs, you lose 20 dollars.
is it possible to adjust the pay amounts to make the game "fair" to both the dealer and the challenger if the expected payoff is 1.093?

Answer 1

sure, charge that much to play.

Answer 2

probability of all head or all tails is 1/32+1/32=2/32
P(1 H) = 5/32
P(2 H) = 10/32
P(3 H)= 10/32
P(4 H)= 5/32
expected value is 1*5/32 +2*10/32 +3*10/32+4*5/32-20*2/32

Answer 3

but question is not clear since obviously you can adjust your payoff in infinite number of ways. numerator of that fraction is 5+20+30+20-40 and when you divide by 32 you get 1.093 rounded. fair would mean the numerator was 0, so perhaps the easiest thing to do it so write 75-2x = 0 and solve for x as what you lose if you get all heads or all tails. this would make the game fair.

Answer 4

in other words instead of losing $20 you would lose $37.5 for all heads or tails. then game would be fair.