A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is k/3¹¹, then k is equal to :

Solution

Maths sir · 4 steps

Find p for one throw first, then add the binomial terms for exactly 4 and exactly 5 successes.

1. One throw. A total of 5 comes from (1,4), (4,1), (2,3), (3,2), which is 4 outcomes out of 36. So p = 4/36 = 1/9 and q = 8/9.

   P(X ≥ 4) = C(5, 4)(1/9)⁴8/9 + C(5, 5)(1/9)⁵

2. Add the terms. The first term is (5 × 8)/9⁵ = 40/9⁵ and the second is 1/9⁵. Together they give 41/9⁵ = 41/3¹⁰.
3. Match the power. The question wants a denominator of 3¹¹, so multiply top and bottom by 3: 41/3¹⁰ = 123/3¹¹.
4. So. k = 123, which is option (c).

The idea to keep. For an "at least" question near the top end, add the last few binomial terms directly. At the end, rewrite the answer over the exact denominator the question gives you.

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