For `P(A)=(3)/(8)`, `P(B)=(1)/(2)`, `P(AuuB)=(5)/(8)` which of the following is/are true ?

Options

A. `P"("A^(C)//B")"=2P"("A//B^(C)")"`
B. `P(B)=P(A//B)`
C. `15P"("A^(C)//B^(C)")"=8P"("B//A^(C)")"`
D. `P"("A//B^(C)")"=(AnnB)`

Answer

A, B, C, D

Explanation

`P(AuuB)=P(A)+P(B)-P(AnnB)`
`or 5/8=3/8+4/8-P(AnnB)`
` or P(AnnB) = 2/8=1/4`
Now, `P(A//B)=(P(A^(c)nnB))/(P(B))`
`=(P(B)-P(AnnB))/(P(B))=1-2((1)/(4))=1/2`
`2P(A//B^(c))=(2P(AnnB^(c)))/(P(B^(c)))`
`=(2(P(A)-P(AnnB)))/(1-P(B))`
`=4((3)/(8)-(2)/(8))=1/2`
Hence, option (1) is correct.
`P(A//B)=(P(annB))/(P(B))=1/4xx2/1=1/2-P(B)`
Hence, option (2) is correct. Again,
`P(A^(c)//B^(c))=(P(A^(c)nnB^(c)))/(P(B^(c)))`
`=(1-P(AuuB))/(1-P(B))`
`=2(1-(5)/(8))=3/4 `
`P(B//A^(c))=(P(BnnA^(c)))/(1-P(A))`
`=(P(B)-P(AnnB))/(5//8)`
`=(1//2-1//4)/(5//8)=1/4xx8/5=2/5`
`8P(A^(c)//B^(c))=15P(B//A^(c))`
Hence, option (3) is not correct. Again,
`2P(A//B^(c))=1/2`
` or P(A//B^(c))=1/4=P(AnnB)`
Hence, option (4) is correct,