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Question

If A and B are two independent events such that P(¯¯¯¯AB)=215 and P(A¯¯¯¯B)=16, then find P(A) and P(B).

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Solution

Given P(¯AB)=215P(¯A)P(B)=215--- (1)

Since events A and B are independent, ¯A and B are also independent.
P(A¯B)=16P(A)P(¯B)=16--- (2)

As events A and B are independent, A, ¯B are also independent.

Let P(A) = x, P(B) = y
P(¯A)=1x and P(¯B)=1y

From (1) and (2), we get
(1x)y=215 i.e., yxy=215--- (3)
and x(1y)=16 i.e., xxy=16 -- (4)

Subtracting (3) and (4), we get
xy=16215xy=130x=y+130 ----(5)

Putting this value of x in (3), we get
y(y+130)y=2152930yy2=215

30y229y+4=0(5y4)(6y1)=0

y=45,16

From (5), when y=45,x=45+130=56
when y=16,x=16+130=15

Hence, if P(A)=56, then P(B)=45
and if P(A)=15, then P(B)=16

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