probability problem 2

**juliak** · September 18th 2010, 06:28 PM

A and B are two independent events such that P(A)=X and P(AUB)=Y, Y>X. Show that:

P(B)=(Y-X)/(1-X)

I don't know how to do this - could some one help me please?

**Soroban** · September 18th 2010, 06:59 PM

Hello, juliak!

$A\text{ and }B\text{ are two independent events such that:}$

. . $P(A)=X\,\text{ and }\,P(A \cup B)=Y,\;\;Y>X$

$\text{Show that: }\;P(B)\:=\:\dfrac{Y-X}{1-X}$

We know this formula: . $P(A \cup B) \;=\;P(A) + P(B) - P(A \cap B)$ .[1]

We are given: . $P(A \cup B) = Y,\;P(A) = X$

Since $\,A$ and $\,B$ are independent: . $P(A \cap B) \:=\:P(A)\!\cdot\!P(B) \;=\;X\!\cdot\!P(B)$

Substitute into [1]: . $Y \;=\;X + P(B) - X\!\cdot\!P(B)$

Then we have: . $P(B) - X\!\cdot\1P(B) \;=\;Y - X$

. . . . . Factor: . . . $P(B)\!\cdot\!(1 - X) \;=\;Y-X$

. . .Therefore: . . . . . . . . . $P(B) \;=\;\dfrac{Y-X}{1-X}$

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