Question

Alice and Bob play the following game. They toss 5 fair coins. If all tosses are Heads, Bob wins. If the number of Heads tosses is zero or one, Alice wins.

Otherwise, they repeat, tossing five coins on each round, until the game is decided.

(a) Compute the expected number of coin tosses needed to decide the game.

(b) Compute the probability that Alice wins

Answer 1

Answer:-

Given That:-

Alice and Bob play the following game. They toss 5 fair coins. If all tosses are Heads, Bob wins. If the number of Heads tosses is zero or one, Alice wins.Otherwise, they repeat, tossing five coins on each round, until the game is decided.

Given,

Let the expected number of toss = E

If no one wins in first five tosses we go for another round.

Probability of bob win in one round = (All heads)

Probability of Alice wins in one round = (4 fails OR 5 tails)

(a) Compute the expected number of coin tosses needed to decide the game.

To calculate expected toss:-

E = 160/3

E = 53.00 tosses

(b) Compute the probability that Alice wins

Probability of Alice win is two times that of Bob

P(A) + P(B) = 1

P(A) + 1/2 P(A) = 1

P(A) = 2/3

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