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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability5 Marks
Question
A and B are two events such that $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{4}.$ Find:
  1. $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$
  2. $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$
  3. $\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)$
  4. $\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)$
✓

Answer

Here, $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{4}$
  1. $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{\text{P}(\text{A}\cap\text{B})}{\text{P}(\text{B})}=\frac{\frac{1}{4}}{\frac{1}{3}}=\frac{3}{4}$
  2. $\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)=\frac{\text{P}(\text{A}\cap\text{B})}{\text{P}(\text{A})}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$
  3. $\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)=\frac{\text{P}(\text{A}'\cap\text{B})}{\text{P}(\text{B})}=\frac{\text{P}(\text{B})-\text{P}(\text{A}\cup\text{B})}{\text{P}(\text{B})}$ $=\frac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{3}}=\frac{\frac{1}{12}}{\frac{1}{3}}=\frac{1}{4}$
  4. $\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)=\frac{\text{P}(\text{A}'\cap\text{B}')}{\text{P}(\text{B})}=\frac{1-\text{P}(\text{A}\cup\text{B})}{1-\text{P}(\text{B})}$
$=\frac{1-\big[\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})\big]}{1-\text{P}(\text{B})}$

$=\frac{1-\big[\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\big]}{1-\frac{1}{3}}=\frac{1-\Big(\frac{5}{6}-\frac{1}{4}\Big)}{\frac{2}{3}}$

$=\frac{1-\frac{14}{24}}{\frac{2}{3}}=\frac{\frac{10}{24}}{\frac{2}{3}}$

$=\frac{30}{48}=\frac{5}{8}$

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