An experiment consists of tossing 3 fair​ (not weighted)​ coins, except one of the 3 coins has a head on both sides. Compute the probability of obtaining less than 3 heads .

Answer 1

Approach 1:

Subtract out the opposite scenario

P(less than 3 heads) = 1 - P(all 3 are heads) = 1 - (1/2)*(1/2)*1 = 3/4

Approach 2:

Calculate it directly

P(less than 3 heads) = P(0 heads) + P(1 head) + P(2 heads)

P(0 heads) = 0. It is impossible since one of the coins is guaranteed to be a head

P(1 head) = 1/2 * 1/2 * 1 = 1/4. This can only happen with Tail, Tail, Head

P(2 heads) = 1/2*1/2*1 + 1/2*1/2*1 = 1/2. This can happen with Tail, Head, Head or Head, Tail, Head

P(less than 3 heads) = 0 + 1/4 + 1/2 = 3/4