Question

2) At the end of the latest video game, Grad School Zombies, you are faced with a series of four boss-level zombies, one at a time. You are a prey good player, so in each battle you have an 80% chance of winning. If you lose a battle, the game is over. Assume that each of the four battles is independent of the others.
a. What is the probability that you defeat all four opponents?
b. What is the probability that you defeat at least two opponents?
c. If you play the game three times, what is the probability that you defeat all four opponents at least once?

Answer 1

P(win), p = 0.8

P(loss), q = 0.2

n = 4

This is a binomial distribution with limited trails and given probability of success and failure

Let X be the random variable which tells the probability of number of success

a) P(X=4)

= C(4,4)*(0.8)^4*(0.2)^0

= 1*0.8^4

= 0.4096

b) P(X=2) + P(X=3) + P(X=4)

This is equivalent to 1-P(X=0)-P(X=1)

P(X=0) = C(4,0)*(0.8)^0*(0.2)^4 = 0.0016

P(X=1) = C(4,1)*(0.8)^1*(0.2)^3 = 0.0256

So, required probability = 1 - 0.0016 - 0.0256 =0.9728

c) This will be indepndent for each game

1-Probability that you will never defeat all 4 opponents in a game

Just substitute P(X=0) and P(X=4) from above and you will get the answer

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