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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability5 Marks
Question
Probability of solving 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
  1. The problem is solved.
  2. Exactly one of them solves the problem.
✓

Answer

Probability of solving the problem by A, P(A) $=\frac{1}{2}$Probability of solving the problem by B, P(B) $=\frac{1}{3}$
Since the problem is solved independently by A and B,
$\therefore\text{P}(\text{AB})=\text{P}(\text{A})\cdot\text{P}(\text{B})=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}$
$\text{P}(\text{A}')=1-\text{P}(\text{A})=1-\frac{1}{2}=\frac{1}{2}$
$\text{P}(\text{B}')=1-\text{P}(\text{B})=1-\frac{1}{3}=\frac{2}{3}$
  1. Probability that the problem is solved $=\text{P}(\text{A}\cup\text{B})$
$=\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{AB})$
$=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}$
$=\frac{4}{6}$
$=\frac{2}{3}$
  1. Probability that exactly one of them solves the problem is given by,
$\text{P}(\text{A}).\text{P}(\text{B}')+\text{P}(\text{B}).\text{P}(\text{A}')$
$ =\frac{1}{2}\times\frac{2}{3}+\frac{1}{2}\times\frac{1}{3}$
$=\frac{1}{3}+\frac{1}{6}$
$=\frac{1}{2}$

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