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Probability — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsProbability2 Marks
Question
The probability that a student will solve problem A is 2/3, and the probability that he will not solve problem B is 5/9. If the probability that student solves at least one problem is 4/5, what is the probability that he will solve both the problems?
✓

Answer

Let event A: student solves problem A
$
\therefore \mathrm{P}(\mathrm{A})=\frac{2}{3}
$
event $B$ : student solves problem $B$.
$\therefore$ event B': student will not solve problem
B.
$
\therefore \mathrm{P}\left(\mathrm{B}^{\prime}\right)=\frac{5}{9}
$
$
\therefore \mathrm{P}(\mathrm{B})=1-\mathrm{P}\left(\mathrm{B}^{\prime}\right)=1-\frac{5}{9}=\frac{4}{9}
$
Probability that student solves at least one
$
\begin{aligned}
& \text { problem }=\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{4}{5} \\
& \therefore \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\
& \therefore \text { required probability }=\mathrm{P} \text { (he will solve both } \\
& \text { the problems) } \\
& =\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \\
& =\frac{2}{3}+\frac{4}{9}-\frac{4}{5}=\frac{14}{45} \\
\end{aligned}
$

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