Question

1. A videogame player has to play five opponents in consecutive order. She has an 80% probability of defeating each of them. Assume the results from opponents are independent and that when the player is defeated the game ends.

a. Describe the outcomes of 1 random process, and then build a probability tree showing what happens if the player defeats the first, second, third, fourth opponent.

b. What is the probability that the player defeats all opponents?

c. What is the probability that the player defeats at least two opponents in a game?

2. A national survey of couples showed that 30% of wives watched “America’s Next Top Model”. For husbands in the sample, this percentage was 50%. Also, if the wife watched, the probability that the husband watched increased to 60%. For a couple drawn at random, what is the probability that:

a. Both watch
b. At least one watches
c. Neither watches
d. If the husband watches, the wife watches
e. If the husband does not watch, the wife watches

Answer 1

Ans 1)

a) Let W be the event that Player 1 wins a game from an opponent. The probability tree for the games with the 1st, 2nd, 3rd and 4th opponent is as follows. Each intermediate node represents a Game. The left branch indicates a Win and the right branch indicates a Loss. The nodes labelled END represent the end of the game by either a loss to one of the opponents in games 1-4, or the player winning all the 4 games.

b) As the results from opponents are independent,

P(defeating all opponents) = P(defeating opponent 1)*P(defeating opponent 2)*P(defeating opponent 3)*P(defeating opponent 4) = 0.8⁴ = 0.41

c) P(defeating at least 2 opponents) = P(defeating 2 opponents) + P(defeating 3 opponents) + P(defeating 4 opponents)

= 0.8*0.8*0.2 + 0.8*0.8*0.8*0.2 + 0.8*0.8*0.8*0.8*0.2 = 0.128 + 0.1024 + 0.08192 = 0.312

Ans 2)

Let's define the events as:

W: Wife watches the show, H: Husband watches the show

Given, P(W) = 0.3, P(H) = 0.5, and P(H/W) = 0.6

a) From the law of conditional probability, P(H/W) = P(H W) / P(W)

P(Both watch) = P(H W) = P(H/W) * P(W) = 0.6*0.3 = 0.18

b) P(At least one watches) = P(H W) = P(H) + P(W) - P(H W) = 0.62

c) P(Neither watches) = 1 - P(At least one watches) = 1 - P(H W) = 1 - 0.62 = 0.38

d) P(if husband watches, then wife watches) = P(W/H) = P(W H) / P(H) = 0.18 / 0.3 = 0.6

e) P(if husband does not watch, wife watches) = P(W / ) = P(W ) / P()

But, P(W ) = P(W) - P(W H) = 0.3 - 0.18 = 0.12 (shaded region shown in the Venn diagram below)

P() = 1 - P(H) = 0.5

P(if husband does not watch, wife watches) = 0.12 / 0.5 = 0.24