Question 9

Mathematics Probability Hard

A problem in Mathematics is given to 3 students A, B and C. If the probability of A solving the problem is&nbsp;<span class="math-tex">\(\dfrac{1}{2}\)</span>&nbsp;and B not solving it is&nbsp;<span class="math-tex">\(\dfrac{1}{4}\)</span>&nbsp;and the whole probability of the problem being solved is <span class="math-tex">\(\dfrac{63}{64}\)</span>, then what is the probability of solving it by C?

(A) <span class="math-tex">\(\dfrac18\)</span>
(B) <span class="math-tex">\(\dfrac1{64}\)</span>
(C) <span class="math-tex">\(\dfrac78\)</span>
(D) <span class="math-tex">\(\dfrac12\)</span>
โœ“ Correct Answer: Option C Verified Solution

๐Ÿ“˜ Step-by-Step Solution & Detailed Explanation:

Probability that B will solve the problem: \[ P(B) = 1 - \frac{1}{4} = \frac{3}{4} \] Let the probability that C will solve the problem be \( p \). The probability that 'none of them' is able to solve the problem is given by: \[ P(\text{None}) = \frac{1}{2} \times \frac{1}{4} \times (1 - p) = \frac{1 - p}{8}. \] From the given information: \[ \frac{1 - p}{8} = \frac{1}{64}. \] Solving for \( p \): \[ 1 - p = \frac{1}{8} \cdot \frac{1}{64} = \frac{1}{8}. \] Therefore: \[ p = \frac{7}{8}. \]

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