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Class 12 Probability · Long Answer3 Marksprobability_distributions

Class 12 Maths · Probability · Long Answer #71

A coin is three times as likely to show a head as a tail. Toss it independently twice. Find the distribution and mean of the number of tails.

✅ Step-by-Step Solution

Final Answer: P(X=0)=9/16; P(X=1)=3/8; P(X=2)=1/16. Mean=1/2.
  1. Let X count successes in 2 independent trials. Each succeeds with probability 1/4; hence X~Binomial(2,1/4).
  2. For k=0,…,2, P(X=k)=C(2,k)(1/4)^k(3/4)^(2−k). Evaluate for each k. The probabilities are P(X=0)=9/16; P(X=1)=3/8; P(X=2)=1/16. Their sum is 1.
  3. E(X)=ΣxP(X=x)=(0)(9/16)+(1)(3/8)+(2)(1/16)=1/2.
  4. P(X=0)=9/16; P(X=1)=3/8; P(X=2)=1/16. Mean=1/2.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let X count successes in 2 independent trials. Each succeeds with probability 1/4; hence X~Binomial(2,1/4).
  • [1 mark] For k=0,…,2, P(X=k)=C(2,k)(1/4)^k(3/4)^(2−k). Evaluate for each k. The probabilities are P(X=0)=9/16; P(X=1)=3/8; P(X=2)=1/16. Their sum is 1.
  • [1 mark] E(X)=ΣxP(X=x)=(0)(9/16)+(1)(3/8)+(2)(1/16)=1/2.
  • [0.5 marks] P(X=0)=9/16; P(X=1)=3/8; P(X=2)=1/16. Mean=1/2.

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