Which of the following statement is/are correct?

Options

A. Three coins are tossed once. At least two of them must land the same way. No matter whether they land heads or tails, the third coin is equally likely to land either the same ways or oppositely. So, the chance that all the three coins land the same ways is `1//2.`
B. `Let 0ltP(B)lt1and P(A|B)=P(A//B^(C)).` Then A and B are independent.
C. Suppose an urn contains "w" white and "b" black balls and a ball is drawn from it and is replaced along with "d" additional balls of the same color. Now a second ball is drawn from it. The proability that the second drawn ball is ehite in independent of the value of "d"
D. A,B,C simultaneously satisfy
`P(ABC)=P(A)P(B)P(C)`
`P(ABbarC)=P(A)P(B)P(barC)`
`P(AbarBC)=P(A)P(barB)P(C)`
`P(A-BC)=P(barA)P(B)P(C)`
Then A, B C are independent.

Answer

B, C, D

Explanation

(1) False.
`P(T T Tor HHH)=1/8+1/8=1/4`
2 `(P(AnnB))/(P(B))=(P(AnnB'))/(1-P(B))=(P(A)-P(AnnB))/(1-P(B))`
`P(AnnB)[1-P(B)]=P(B)P(A)-P(B)P(AnnB)`
`P(AnnB)=P(A)P(B)`
Hence, the given statement is true.
(3) Let `E_(1)` be the white ball drawn in first draw, `E_(2)` be the event the black ball is drawn in second draw, E be the event that white ball is drawn in second draw. Then
`P(E)=P(E//E_(1))P(E_(1))+P(E//E_(2))P(E_(2))`
`=(d+w)/(w+b+d)((w)/(a+b))+(w)/(w+b+d)((b)/(w+b))`
`=((w)/(w+b))((d+w)/(w+b+d)+(b)/(w+b+d))=((w)/(w+b))`
which is independent of d.
(4) To prove that A,B, C are pairwase independent only. Now,
`P(AnnB)=P((AnnBnnC)uu(AnnBnnC)`
`=P(AnnBnnC)+P(A\nnBnnC)`
`=P(A)P(B)P(barC)+P(A)P(B)P(C)("given")`
`=P(A)xxP(B)[P(C')+P(barC)]`
`=P(A)xxP(B)`
Similarly, for the other two. Hence, this statement is correct.