`A` and `B` are two events defined as follows:
A:It rains today with`P(A)=40 %`
B: It rains tomorrow with `P(B)=50 %`
Also, `P` (it rains today and tomorrow) `=30 %`
Also, `E_1: P(AnnB)//(AuuB)` and `E_2: P({(Ann barB )or(Bnn barA )}//(AuuB))` . Then which of the following is/are true ?
(a) `A` and `B` are independent events
(b) `P(A//B)`<`P(B//A)`
(c)`E_1` and `E_2` are equiprobable
(d)`P(A/(AuuB))=P(B//(AuuB))`

Options

A. `P(A//B)ltP(B//A)`
B. `P(A//B)ltP(B//A)`
C. `E_(1)and E_(2)` are equiprobable
D. `P(A//(AuuB))=P(B//(AuuB))`

Answer

B, C

Explanation

`P(A)=0.4,P(B)=0.5,P(AnnB)=0.3`
`impliesP(AuuB)=P(A)+P(B)-P(AnnB)=0.6`
Now, `P(E_(1))=(P(AnnB))/(P(AuuB))=(0.3)/(0.6)=1/2`
`P(E_(2))=(P(AnnbarB)+P(barAnnB))/(P(AnnB))=(0.1+0.2)/(0.6)=1/2`
`P(A//B)(P(AnnB))/(P(B))=(0.3)/(0.5)=3/5=0.60,`
`P(B//A)=(0.3)/(0.4)=(3)/(4)=0.75`
`P(A//(AuuB))=(P(A))/(P(AuuB))=(0.4)/(0.6)=2/3`
`(P(B))/(P(AuuB))=(0.5)/(0.6)=5/6`