A problem in Mathematics is given to 3 students A, B and C. If the probability of A solving the problem is <span class="math-tex">\(\dfrac{1}{2}\)</span> and B not solving it is <span class="math-tex">\(\dfrac{1}{4}\)</span> 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?
Step-by-step Solution:
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: \[ p = \frac{7}{8}. \]