A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

Solution

Maths sir · 4 steps

First turn the bias into numbers, P(H) = 2/3 and P(T) = 1/3, then count the orders.

1. Find the probabilities. Let P(T) = p. A head is twice as likely, so P(H) = 2p. They must add to 1, so 3p = 1, which gives P(T) = 1/3 and P(H) = 2/3.
2. One fixed order. The tosses are independent, so the order TTH has probability 1/3 × 1/3 × 2/3 = 2/27.
3. Count the orders. The single head can come on toss 1, 2 or 3: HTT, THT, TTH. Each order has the same probability 2/27, so multiply by 3.

   P(2 tails, 1 head) = 3 × (1/3)² × 2/3 = 6/27 = 2/9

4. So. The probability of two tails and one head is 2/9, which is option (b).

The idea to keep. For a biased coin, find P(H) and P(T) from the ratio first. Then probability of one order times the number of orders, C(n, r)pʳ qⁿ⁻ʳ.

Answer from the NTA final answer key. Solution by PrepWiser.