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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability4 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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