Question

Sally plays a game and wins with probability p. Every week, she plays until she wins two games, and then stops for the week. Sally calls it a "lucky week" if she manages to achieve her goal in 7 or less games.

a) If p = 0.2, what's the probability that Sally will have a "lucky week" next week?

b) What's the probability of exactly 3 "lucky weeks" in the next 5 weeks? What's the expected number of "lucky weeks" Sally will have in the next 10 weeks?

c) Let X be the number of games Sally plays in a week, and let q = 1 – p. Find the expectation E[X].

d) If Sally pays $1 to play each game, and gets $5 for each game she wins, what's her expected earning at the end of each week?

Answer 1

a)

Sally achieve in 2 games = P ( X =    2   ) = C(2,2) * 0.2^2 * (1-0.2)^0 =            0.040

Sally achieve in 3 games = P ( X =    2  ) = C(2,1) * 0.2^1 * (1-0.2)^1 * 0.2 = 0.064

Sally achieve in 4 games = P ( X =    2   ) = 0.0768

Sally achieve in 5 games = P ( X =    2  ) = 0.0819

Sally achieve in 6 games = P ( X =    2   ) =0.0819

Sally achieve in 7 games = P ( X =    2   ) = 0.0786

Probability = 0.4232

b)

Exactly 3 lucky weeks in 5 weeks =

P ( X =    3   ) = C(5,3) * 0.4232^3 * (1-0.4232)^2 =            0.252

Expecetd number of lucky weeks in 10 weeks = np = 10 *0.4232 = 4.232

c)

Since we know

Mean = np

here mean = 2(games to win)

np =2

n = 2/0.2 = 10

d)

Expecetd games = 10
she pays $1 and gets $5 for each game she wins

expected earning =  (50 - 10) = 40

Please revert back in case of any doubt.

Please upvote. Thanks in advance.