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Let A and B be independent events with P(A)=0.3 and P(B)=0.4. Find (i) P(A \cap B)
(ii) P(A \cup B)
(iii) P(A \mid B)
(iv) P(B \mid A)
CBSE | Maths | NCERT | Probability
Let A and B be independent events with P(A)=0.3 and P(B)=0.4. Find (i) P(A \cap B)
(ii) P(A \cup B)
(iii) P(A \mid B)
(iv) P(B \mid A)

Solution:

Given: \mathrm{P}(\mathrm{A})=0.3 and \mathrm{P}(\mathrm{B})=0.4

(i) P(A \cap B)
Given P ( A )=0.3 and P ( B )=0.4

When A and B are independent.Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

    \[\begin{aligned}&\Rightarrow P(A \cap B)=P(A) \cdot P(B) \&\Rightarrow P(A \cap B)=0.3 \times 0.4 \&\Rightarrow P(A \cap B)=0.12\end{aligned}\]


(ii) P(A \cup B)
As we know, P(A \cup B)=P(A)+P(B)-P(A \cap B)

    \[\Rightarrow P(A \cup B)=0.3+0.4-0.12\]


    \[\Rightarrow P(A \cup B)=0.58\]


(iii) P ( A \mid B )
As we know P(A \mid B)=\frac{P(A \cap B)}{P(B)}

    \[\begin{aligned}&\Rightarrow P(A \mid B)=\frac{0.12}{0.4} \&\Rightarrow P(A \mid B)=0.3\end{aligned}\]


(iv) P ( B \mid A )
As we know P(B \mid A)=\frac{P(A \cap B)}{P(A)}

    \[\begin{aligned}&\Rightarrow P(B \mid A)=\frac{0.12}{0.3} \&\Rightarrow P(B \mid A)=0.4\end{aligned}\]

(iii) \mathrm{P}(\mathrm{A} \mid \mathrm{B})

    \[\text { As we know } \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}\]

\Rightarrow \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{0.12}{0.4}

    \[\Rightarrow \mathrm{P}(\mathrm{A} \mid \mathrm{B})=0.3\]

As we know the formula, \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}

\Rightarrow \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{0.12}{0.3}

\Rightarrow \mathrm{P}(\mathrm{B} \mid \mathrm{A})=0.4

i

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