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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability5 Marks
Question
Probabilities of solving a specific problem independently by A and B are $\frac{1}{2} \text{and}\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that(i) the problem is solved (ii) exactly one of them solves the problem.
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Answer

Here P(A) = $\frac{1}{2}$    $\therefore$ P(Not A) = $\frac{1}{2}$
and P(B) = $\frac{1}{3}$      $\therefore$ P(Not A) = $\frac{2}{3}$
  1.  P(problem is solved) = 1 - P (problem is not solved)
= 1 - P(Not A) . P(Not B)
=$1-\frac{1}{2}\cdot\frac{2}{3}=\frac{2}{3}$
  1. P(exactly one of them solves) =$\text{P(A)}\cdot\text{P}\overline{(B)}+\text{P(B)}\cdot\text{P}\overline{(A)}$
$=\frac{1}{2}\cdot\frac{2}{3}+\frac{1}{3}\cdot\frac{1}{2}$
$=\frac{1}{2}$.

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