Question

You have a competition with your friend to see who can get the first free-shot into the basket. Your free-throw percentage is 0.25, whereas your friend’s free-throw percentage is 0.33.

a) On average, how many free-throw attempts do you need to make in order to achieve the first free-throw in the basket?

b) On average, how many free-throw attempts does your friend need to make in order to achieve the first free-throw in the basket?

c) Compute the probability that the bet ends at or before you both make 10 free-throw attempts.

d) Compute the probability of a tie, assuming that the game ends at or before 10 free-throw attempts.

e) Compute the probability that you win the bet, assuming that the game ends at or before 10 free-throw attempts.

f) Assume that you will pay your friend $1, if he wins, $0 in the case of a tie, and he pays you $x if you win the bet. Compute the minimum $x you would accept for this bet to be profitable in the long run?

Recall from class that P(X>n)=(1-p)n when X~Geom(p).

Answer 1

a)

Let X and Y be the number of free-throw attempts needed by you and your friend respectively.

X ~ Geom(0.25) Y ~ Geom(0.33)

Average free-throw attempts you need to make = 1/p = 1/0.25 = 4

b)

Average free-throw attempts the friend need to make = 1/p = 1/0.33 = 3.03

c)

Probability that none of you and your friend can make the shot = (1 - 0.25) * (1 - 0.33) = 0.5025

Probability that any of you or your friend can make the shot = 1 - 0.5025 = 0.4975

Let Z be the number of free-throw attempts needed such that anyone can make the shot.

Probability that the bet ends at or before you both make 10 free-throw attempts = P(Z 10)

= 1 - P(Z > 10) = 1 - (1 - 0.4975)¹⁰ = 0.9989735

d)

Probability of a tie = Probability that both can make the shot at one time = 0.25 * 0.33 = 0.0825

Probability of a tie at n throw = 0.5025^n-1 * 0.0825

Probability of a tie, assuming that the game ends at or before 10 free-throw attempts

= 0.0825 * (1 - 0.5025⁹) / (1 - 0.5025)

= 0.1654904

e)

Probability that you win and friend does not make the throw  = 0.25 * (1 - 0.33) = 0.1675

Probability of a tie at n throw = 0.5025^n-1 * 0.1675

Probability of a tie, assuming that the game ends at or before 10 free-throw attempts

= 0.1675 * (1 - 0.5025⁹) / (1 - 0.5025)

= 0.3359956

f)

Probability of your friends win = 0.9989735 - 0.1654904 - 0.3359956 = 0.4974875

Expected gain = 0.4974875 * (-1) + 0.3359956 * x

Value of x for Expected gain > 0

0.4974875 * (-1) + 0.3359956 * x > 0

x > 1.48

Minimum $x for be profitable in the long run = $1.48