Class 12 Probability · Long Answer3 Marksprobability_distributions

Class 12 Maths · Probability · Long Answer #66

Select 3 distinct numbers uniformly from {1,2,3,4,5,6}. X is their minimum. Find its distribution and mean.

✅ Step-by-Step Solution

Final Answer: P(X=1)=1/2; P(X=2)=3/10; P(X=3)=3/20; P(X=4)=1/20. Mean=7/4.

There are C(6,3)=20 equally likely subsets; the minimum can be 1,2,3,4.
For minimum k include k and choose the other two from {k+1,…,6}; the counts C(6−k,2) are 10,6,3,1. The probabilities are P(X=1)=1/2; P(X=2)=3/10; P(X=3)=3/20; P(X=4)=1/20. Their sum is 1.
E(X)=ΣxP(X=x)=(1)(1/2)+(2)(3/10)+(3)(3/20)+(4)(1/20)=7/4.
P(X=1)=1/2; P(X=2)=3/10; P(X=3)=3/20; P(X=4)=1/20. Mean=7/4.

[0.5 marks] There are C(6,3)=20 equally likely subsets; the minimum can be 1,2,3,4.
[1 mark] For minimum k include k and choose the other two from {k+1,…,6}; the counts C(6−k,2) are 10,6,3,1. The probabilities are P(X=1)=1/2; P(X=2)=3/10; P(X=3)=3/20; P(X=4)=1/20. Their sum is 1.
[1 mark] E(X)=ΣxP(X=x)=(1)(1/2)+(2)(3/10)+(3)(3/20)+(4)(1/20)=7/4.
[0.5 marks] P(X=1)=1/2; P(X=2)=3/10; P(X=3)=3/20; P(X=4)=1/20. Mean=7/4.

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