# Suppose that Pr(B)= 2Pr(A) and that Pr(A U B)= 0.75. Find Pr(A) if: (a) A and B are mutually exclusive. (b) A and B are independent.

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### 2 Answers

In general case `Pr(A cup B)=Pr(A)+Pr(B)+Pr(A cap B)`** (1)**

**(a)** If A and B are mutually exclusive then `Pr(A cap B)=0` hence `Pr(A cup B)=Pr(A)+Pr(B)` and since `Pr(B)=2Pr(A)` we have system of two equations with two unknowns:

`Pr(B)=2Pr(A)`

`Pr(A)+Pr(B)=0.75`

Now in second equation instead of `Pr(B)` we put `2Pr(A)`.

`Pr(A)+2Pr(A)=0.75 => Pr(A)=0.25`

`Pr(B)=2cdot0.25=0.5`

**(b)**** **If A and B are independent, then `Pr(A cap B)=Pr(A)Pr(B)` and since `Pr(B)=2Pr(A)` we have

`Pr(A cap B)=Pr(A)cdot 2Pr(A)=2Pr^2(A)`

Now from **(1)** we have:

`0.75=Pr(A)+Pr(B)+ 2Pr^2(A)`

`0.75= Pr(A)+2Pr(A)+2Pr^2(A)`

So for `x=Pr(A)` we have quadratic equation:

`2x^2+3x-0.75=0`

`x_(1,2)=(-3pm sqrt(15))/4`

Now since probability can't be negative our only solution is ` `

`Pr(A)=(-3+sqrt(15))/4approx0.218246`

`Pr(B)=(-3+sqrt(15))/2approx0.436492`

Thank you very much. I appreciate it.