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  • Solve this problem - Statistics and Probability - JEE Main-4

Let A\; \; and\; \; B be two events such that P(\overline{A\cup B})=\frac{1}{6},\; \; \; P(A\cap B)=\frac{1}{4}\; \; and\; \; P(\bar{A})=\frac{1}{4},  where  \bar{A} stands for complement of event A. Then events A\; \; and\; \; B are

  • Option 1)

    equally likely but not independent

  • Option 2)

    equally likely and mutually exclusive

  • Option 3)

    mutually exclusive and independent

  • Option 4)

    independent but not equally likely

 

Answers (1)

best_answer

As we have learnt,

 

Complementary events -

Let S be the sample space  for a random experiment and let E be the event then complement of event E is 

S-n\left ( E \right )= \left n( E{}' \right )

\therefore n\left ( E \right ) + n\left ( E{}' \right )= n\left ( S \right )

- wherein

where E' contains those element which E does not contain.

 

 

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 )

-

 

 

Multiplication Theorem of Probability -

If A and B are any two events then 

P\left ( A\cap B \right )= P\left ( B \right )\cdot P\left ( \frac{A}{B} \right )


 

- wherein

where B\neq \phi

 

 We have,

\\P(A\cup B) = 1 - P(\overline{A\cup B}) = \frac{5}{6} \\ P(A\cap B)= \frac{1}{4}

P(A) = 1 - P(\overline{A}) = \frac{3}{4}

Now,

\\P(A\cup B) = P(A) + P(B) - P(A\cap B) \\\frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4} \\P(B) = \frac{1}{3}

Now,

\\P(A\cup B) = \frac{1}{4} = \frac{3}{4}\times\frac{1}{3}

\\P(A)\cdot P(B)

Therefore, Events A & B are independent but \\P(A)\neq P(B); So, not equally likely.

 

 

 


Option 1)

equally likely but not independent

Option 2)

equally likely and mutually exclusive

Option 3)

mutually exclusive and independent

Option 4)

independent but not equally likely

Posted by

gaurav

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