Mathematics - Probability Question with Solution | TestHub

Given:

$P(\overline{A})=0.3$

$P(B)=0.4$

$P(A\cap \overline{B})=0.5$

We need to find $P(B/(A\cup \overline{B}))$.

Step-by-step solution:

1. Find $P(A)$:

$P(A)=1-P(\overline{A})=1-0.3=0.7$

2. Find $P(A\cap B)$:

We know that $P(A\cap \overline{B})=P(A)-P(A\cap B)$.

So, $0.5=0.7-P(A\cap B)$.

$P(A\cap B)=0.7-0.5=0.2$.

3. Find $P(B\cap (A\cup \overline{B}))$:

Using the distributive law for sets, $B\cap (A\cup \overline{B})=(B\cap A)\cup (B\cap \overline{B})$.

Since $B\cap \overline{B}=\varnothing$, then $(B\cap A)\cup (B\cap \overline{B})=B\cap A$.

So, $P(B\cap (A\cup \overline{B}))=P(A\cap B)=0.2$.

4. Find $P(A\cup \overline{B})$:

We know that $P(A\cup \overline{B})=P(A)+P(\overline{B})-P(A\cap \overline{B})$.

First, find $P(\overline{B})$:

$P(\overline{B})=1-P(B)=1-0.4=0.6$.

Now, substitute the values:

$P(A\cup \overline{B})=0.7+0.6-0.5=1.3-0.5=0.8$.

5. Calculate $P(B/(A\cup \overline{B}))$:

Using the formula for conditional probability:

$$P(B/(A\cup \overline{B}))=\frac{P(B\cap (A\cup \overline{B}))}{P(A\cup \overline{B})}$$

Substitute the values found in steps 3 and 4:

$$P(B/(A\cup \overline{B}))=\frac{0.2}{0.8}=\frac{1}{4}$$

The final answer is $1/4$.