If A and B are two independent events such that P(A)=\frac{3}{10}  and P(A\cup B)=\frac{4}{5}, then P(A\cap B) is equal to :

 

  • Option 1)

    \frac{3}{35}

  • Option 2)

    \frac{1}{5}

  • Option 3)

    \frac{1}{10}

  • Option 4)

    \frac{3}{14}

 

Answers (1)

As we learnt in

 

Addition Theorem of Probability -

P\left ( A\cup B \right )= P\left ( A \right )+P\left ( B \right )-P\left ( A\cap B \right )

in general:

P\left ( A_{1}\cup A_{2}\cup A_{3}\cdots A_{n} \right )=\sum_{i=1 }^{n}P\left ( A_{i} \right )-\sum_{i< j}^{n}P\left ( A_{i}\cap A_{j} \right )+\sum_{i< j< k}^{n} P\left ( A_{i}\cap A_{j}\cap A_{k} \right )-\cdots -\left ( -1 \right )^{n-1}P\left ( A_{1}\cap A_{2}\cap A_{3}\cdots \cap A_{n} \right )

-

 

 

and

 

P(A)=\frac{3}{10} \: \: let \: P(B)=x

P(A\cap B)=P(A).P(B) \: \: independent \: event

P(A\cup B)= \frac{3}{10}+x-\frac{3}{10}x= \frac{4}{5}

                  \frac{7}{10}x= \frac{1}{2}

                   x= \frac{5}{7}

P(A\cap B)=\frac{3}{10}\times \frac{5}{7}= \frac{3}{14}

                 


Option 1)

\frac{3}{35}

This option is incorrect

Option 2)

\frac{1}{5}

This option is incorrect

Option 3)

\frac{1}{10}

This option is incorrect

Option 4)

\frac{3}{14}

This option is correct

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