Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I need to determine x given the following conditions.

A and B are two events of Ω.

P(not A) = 3x

P(B) = 1/2

P(A or B) = 9x

P(A and B) = 3x

Here's what I thought of (but I know is wrong):

P(A or B) = 9x
P(A) + P(B) = 9x
1 - P(not A) + 1/2 = 9x
1 - 3x + 1/2 = 9x
x = 1/8

The answer is 0.1, but I can't get to it. I can't do the above because P(A or B) = P(A) + P(B) can only be done if P(A and B) = 0, and we are not told that here.

Any ideas? Thank you in advance.

share|cite|improve this question
With $P(1-A)$ you mean $P(\text{not } A)$, I presume? – Lord_Farin Oct 20 '12 at 12:36
Indeed, will edit it. – David Gomes Oct 20 '12 at 12:40
$P(A\cup B) = 9x$ doesn't imply $P(A) + P(B) = 9x$ when $A\cap B\ne\emptyset$. – Frenzy Li Oct 20 '12 at 12:51
Really @FrenzYDT.? Damn, I was pretty sure about that. – David Gomes Oct 20 '12 at 13:03

You need to use that $A = (A \cup B)\setminus B \cup (A \cap B)$. Since $B \subset A \cup B$ and $(A \cup B)\setminus B \cap (A \cap B) = \emptyset$, you have $$ P(A) = P(A \cup B) - P(B) + P(A \cap B) $$ and get $$ 1 - 3x = 9x - \frac{1}{2} + 3x $$ which yields $x = \frac{1}{10}$.

share|cite|improve this answer
Simpler than what I was preparing with conditional expectations; nice. – Lord_Farin Oct 20 '12 at 12:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.