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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability1 Mark
Question
Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that the problem is solved.
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Answer

Given:
P(A) = Probability of solving the problem by A = $\frac{1}{2}$
P(B) = Probability of solving the problem by B = $\frac{1}{3}$
Since, A and B both are independent.
$\Rightarrow$ P(A $\cap$ B) = P(A).P(B)
$\Rightarrow$ P (A $\cap$ B) = $\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$ 
The problem is solved, i.e. it is either solved by A or it is solved by B.
= P(A $\cup$ B)
As we know, P (A $\cup$ B) = P(A) + P(B) - P (A $\cap$ B)
⇒ P (A $\cup$ B) = $\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{4}{6}$ 
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{2}{3}$

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