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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability3 Marks
Question
Three events A, B and C have probabilities $\frac{2}{5},\frac{1}{3}$ and $\frac{1}{2},$ respectively. Given than $\text{P}(\text{A}\cap\text{C})=\frac{1}{5}$ and $\text{P}(\text{B}\cap\text{C})=\frac{1}{4},$ find the values of $\text{P}\Big(\frac{\text{C}}{\text{B}}\Big)$ and $\text{P}(\text{A}'\cap\text{C}').$
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Answer

Here, $\text{P}(\text{A})=\frac{2}{5},\text{P}(\text{B})=\frac{1}{3},\text{P}(\text{C})=\frac{1}{2},\text{P}(\text{B}\cap\text{C})=\frac{1}{4}$ and $\text{P}(\text{B}\cap\text{C})=\frac{1}{4}$
$\therefore\text{P}\Big(\frac{\text{C}}{\text{B}}\Big)=\frac{\text{P}(\text{B}\cap\text{C})}{\text{P}(\text{B})}=\frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}$
And $\text{P}(\text{A}'\cap\text{C}')=1-\text{P}(\text{A}\cup\text{C})$
$=1-\big[\text{P}(\text{A})+\text{P}(\text{C})-\text{P}(\text{A}\cap\text{C})\big]$
$=1-\Big[\frac{2}{5}+\frac{1}{2}-\frac{1}{5}\Big]=1-\Big[\frac{4+5-2}{10}\Big]$
$=1-\frac{7}{10}=\frac{3}{10}$

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