In a certain game, a coin is flipped 4 times. If exactly one head comes up then

Question

In a certain game, a coin is flipped 4 times. If exactly one head comes up then the player gets a $1 profit ($1 more than the cost of the ticket). If exactly 2 heads come up then the player gets a $2 profit. If exactly 3 heads come up then the player gets a $4 profit. If all 4 are heads then the player gets an $8 profit. If no heads come up then he or she loses the cost P of the ticket. What should the cost P of the ticket be to make it a fair game?
The cost should be $. In a certain game, a coin is flipped 4 times. If exactly one head comes up then the player gets a $1 profit ($1 more than the cost of the ticket). If exactly 2 heads come up then the player gets a $2 profit. If exactly 3 heads come up then the player gets a $4 profit. If all 4 are heads then the player gets an $8 profit. If no heads come up then he or she loses the cost P of the ticket. What should the cost P of the ticket be to make it a fair game?
The cost should be $. In a certain game, a coin is flipped 4 times. If exactly one head comes up then the player gets a $1 profit ($1 more than the cost of the ticket). If exactly 2 heads come up then the player gets a $2 profit. If exactly 3 heads come up then the player gets a $4 profit. If all 4 are heads then the player gets an $8 profit. If no heads come up then he or she loses the cost P of the ticket. What should the cost P of the ticket be to make it a fair game?
The cost should be $.

Explanation / Answer

First we calculate the expected profit of the game.

This is a binomial distribution with the following parameters:

n = 4, p = 0.5

Let X denote the number of times head comes up.

P(X=0) = 4C0*(0.5^0)*(0.5^(4-0)) = 0.0625

P(X=1) = 4C1*(0.5^1)*(0.5^(4-1)) = 0.25

P(X=2) = 4C2*(0.5^2)*(0.5^(4-2)) = 0.375

P(X=3) = 4C3*(0.5^3)*(0.5^(4-3)) = 0.25

P(X=4) = 4C4*(0.5^4)*(0.5^(4-4)) = 0.0625

So,

Expected profit = (-P)*P(X=0) + 1*P(X=1) + 2*P(X=2) + 4*P(X=3) + 8*P(X=4) = -P*0.0625 + 1*0.25 + 2*0.375 +

4*0.25 + 8*0.0625 = 2.5 - P*0.0625

In order to make this a fair game, the net profit should be zero,

So,

2.5 - P*0.0625 = 0

Thus,

P = $40

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---