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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability3 Marks
Question
A fair die is rolled. Consider events E = $\{1,\ 3,\ 5\},\ \text{F}=\{2,\ 3\}\ \text{and}\ \text{G}=\{2,\ 3,\ 4,\ 5\}.\ \text{Find}:$ $\text{P}\Big[(\text{E}\cup\text{F})|\text{G}\Big]\ \text{and}\ \text{P}\Big[(\text{E}\cap\text{F})|\text{G}\Big]$
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Answer

$ \text{P}\left(\text{G}\right)=\frac{\text{n}\left(\text{G}\right)}{\text{n}\left(\text{S}\right)}=\frac{4}{6}$
$\text{E}\cup\text{F}=\left(1,\ 2,\ 3,\ 5\right)\ \ \ \ \text{and}\ \text{G}\ \left(2,\ 3,\ 4,\ 5\right)$
$\left(\text{E}\cup\text{F}\right)\cap\text{G}=\left(2,\ 3,\ 5\right)\ \Rightarrow\ \ \ \ \text{n}\left[\left(\text{E}\cup\text{F}\right)\cap\text{G}\right]=3$
$\text{P}\left[\left(\text{E}\ \cup\ \text{F}\right)\cap\text{G}\right]=\frac{3}{6}$
$\text{P}\left(\text{E}\cup\text{F}|\text{G}\right)=\frac{\text{P}\left[\left(\text{E}\ \cup\ \text{F}\right)\cap\ \text{G}\right]}{\text{P}\left(\text{G}\right)}=\frac{\frac{3}{6}}{\frac{4}{6}}=\frac{3}{4}$
$\text{Again}\ \ \ \ \text{E}\cap\text{F}=\left(3\right)$
$\left(\text{E}\cap\text{F}\right)\cap\text{G}=(3)\ \Rightarrow\ \ \ \ \ \text{n}\big[\left(\text{E}\cap\text{F}\right)\cap\text{G}\big]=1$
$\text{P}\left[\left(\text{E}\ \cap\ \text{F}\right)\cap\text{G}\right]=\frac{1}{6}$
$\text{P}\left(\text{E}\cap\text{F}|\text{G}\right)=\frac{\text{P}\big[\left(\text{E}\ \cap\ \text{F}\right)\cap\ \text{G}\big]}{\text{P}\left(\text{G}\right)}=\frac{\frac{1}{6}}{\frac{4}{6}}=\frac{1}{4}$

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