Class 12 Probability · Long Answer3 Marksprobability_distributions

Class 12 Maths · Probability · Long Answer #67

A fair die is rolled at most three times independently. Each score greater than 4 wins ₹5 and ends play; each other score loses ₹1. Find the expected total net gain.

✅ Step-by-Step Solution

Final Answer: P(X=5)=1/3; P(X=4)=2/9; P(X=3)=4/27; P(X=-3)=8/27. Mean=19/9.

A successful roll has probability 1/3 and a failed roll 2/3. Net gains are 5 for first-roll success, 4 for second, 3 for third, or −3 if all fail.
The four disjoint event probabilities are 1/3, (2/3)(1/3)=2/9, (2/3)²(1/3)=4/27, and (2/3)³=8/27. The probabilities are P(X=5)=1/3; P(X=4)=2/9; P(X=3)=4/27; P(X=-3)=8/27. Their sum is 1.
E(X)=ΣxP(X=x)=(5)(1/3)+(4)(2/9)+(3)(4/27)+(-3)(8/27)=19/9.
P(X=5)=1/3; P(X=4)=2/9; P(X=3)=4/27; P(X=-3)=8/27. Mean=19/9.

📋 CBSE Stepwise Marking Scheme

[0.5 marks] A successful roll has probability 1/3 and a failed roll 2/3. Net gains are 5 for first-roll success, 4 for second, 3 for third, or −3 if all fail.
[1 mark] The four disjoint event probabilities are 1/3, (2/3)(1/3)=2/9, (2/3)²(1/3)=4/27, and (2/3)³=8/27. The probabilities are P(X=5)=1/3; P(X=4)=2/9; P(X=3)=4/27; P(X=-3)=8/27. Their sum is 1.
[1 mark] E(X)=ΣxP(X=x)=(5)(1/3)+(4)(2/9)+(3)(4/27)+(-3)(8/27)=19/9.
[0.5 marks] P(X=5)=1/3; P(X=4)=2/9; P(X=3)=4/27; P(X=-3)=8/27. Mean=19/9.

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Class 12 Maths · Probability · Long Answer #67

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 13: Probability

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