Question

Consider this game: each person flips a coin once, if it is heads, the person gets $20, and if it is tails, the person loses $10.

For a group of 5, each person flips a coin once and plays the game described above. What is the probability that this group of 5 will lose money as a group? (Hint: list all possible outcomes for this group and sum up the probabilities of outcomes with a total loss.)

Answer 1

Probability of heads = Probability that the person wins = 0.5

Let X be the number of persons who win. Then X ~ Binomial(n = 5, p = 0.5)

Total money earned by the group = 20X - 10(5 - X)

= 20X - 50 + 10X

= 30X - 50

If total money earned by the group is less than 0, then

30X - 50 < 0

=> 30X < 50

=> X < 5/3

=> X < 1.67

Probability that this group of 5 will lose money = Probability that total money earned by the group is less than 0

= P(X < 1.67)

= P(X = 0) + P(X = 1)

= ⁵C₀  * 0.5⁰ * (1 - 0.5)⁵ +  ⁵C₁  * 0.5¹ * (1 - 0.5)⁴

= 0.5⁵  + 5 * 0.5⁵

= 0.1875