Choose the correct answer from the given four options. Let A and B be two events such that P(A)=3 8 ,P( B )=5 8 and P(A )=3 4 .Then P ( A B ) ( A' B ) is equal to: 1. 2 5 2. 3 8 3. 3 20 4. 6 25

4. 6 25 Solution: We have, P(A)=3 8 ,P( B )=5 8 and P(A )=3 4 Now P(A )=P(A)+P(B)-P(A ) (A )=3 8 +5 8 -3 4 =1 4 ( A B )= P(A ) P(B) = 1 4 5 8 =2 5 and P ( A' B )= P(A' ) P(B) = P(B)-P(A ) P(B) = 5 8 -1 4 5 8 =3 5 ( A B ) ( A' B ) =2 5 3 5 =6 25

Choose the correct answer from the given four options. Let A and …

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Choose the correct answer from the given four options.
Let A and B be two events such that $\text{P}(\text{A})=\frac{3}{8},\text{P}({\text{B}})=\frac{5}{8}$ and $\text{P}(\text{A}\cup\text{B})=\frac{3}{4}.$Then $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)\cdot\text{P}\Big(\frac{\text{A'}}{\text{B}}\Big)$ is equal to:

$\frac{2}{5}$
$\frac{3}{8}$
$\frac{3}{20}$
$\frac{6}{25}$

✓

Answer

$\frac{6}{25}$

Solution:
We have, $\text{P}(\text{A})=\frac{3}{8},\text{P}({\text{B}})=\frac{5}{8}$ and $\text{P}(\text{A}\cup\text{B})=\frac{3}{4}$
Now $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})$
$\Rightarrow\text{P}(\text{A}\cap\text{B})=\frac{3}{8}+\frac{5}{8}-\frac{3}{4}=\frac{1}{4}$
$\because\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{\text{P}(\text{A}\cap\text{B})}{\text{P}(\text{B})}$
$=\frac{\frac{1}{4}}{\frac{5}{8}}=\frac{2}{5}$
and $\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)=\frac{\text{P}(\text{A}'\cap\text{B})}{\text{P}(\text{B})}=\frac{\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})}{\text{P}(\text{B})}$
$=\frac{\frac{5}{8}-\frac{1}{4}}{\frac{5}{8}}=\frac{3}{5}$
$\therefore\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)\cdot\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)$
$=\frac{2}{5}\cdot\frac{3}{5}=\frac{6}{25}$

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