Question

A fair coin is tossed 4 times.

What is the probability of getting exactly 3 heads conditioned on the event that the first two tosses came out the same?

Answer 1

Bayes' Theorem: P(A | B) = P(A & B) / P(B)

P(exactly 3 heads | first two tosses came out the same) = P(first two tosses are the same and exactly 3 heads) / P(first two tosses are the same)

If first two tosses are tails, then we cannot obtain 3 heads on 4 tosses. therefore, P(first two tosses are the same and exactly 3 heads) = P(first two tosses are heads and there in exactly one head in the next two tosses)

= 0.5² x 2x0.5² (HHTH, HHHT)

= 0.125

P(first two tosses are the same) = P(HH) + P(TT)

= 0.5² + 0.5²

= 0.5

P(exactly 3 heads | first two tosses came out the same) = 0.125 / 0.5

= 0.25