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If E(1) and E(2) are two events such tha...

If `E_(1)` and `E_(2)` are two events such that `P(E_(1))=1//4`, `P(E_(2)//E_(1))=1//2` and `P(E_(1)//E_(2))=1//4`, then

A

then `E_(1)` and `E_(2)` are independent

B

`E_(1)` and `E_(2)` are exhaustive

C

`E_(2)` is twice as likely to occur as `E_(1)`

D

Probabilites of the events `E_(1) nn E_(2)`, `E_(1)` and `E_(2)` are in `G.P.`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
`(a,c,d)` `P(E_(2)//E_(1))=(P(E_(1)nnE_(2)))/(P(E_(1)))`
`(1)/(2)=(P(E_(1)nnE_(2)))/(1//4)`
`impliesP(E_(1)nnE_(2))=(1)/(8)=P(E_(2))*P(E_(1)//E_(2))`
`=P(E_(2))*(1)/(4)`
`impliesP(E_(2))=(1)/(2)`
Since `P(E_(1) nnE_(2))=(1)/(8)=P(E_(1))*P(E_(1))*P(E_(2))`, events are independent
Also `P(E_(1)uuE_(2))=(1)/(2)+(1)/(4)-(1)/(8)=(5)/(8)`
`impliesE_(1)` and `E_(2)` are not exhaustive.
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