# Thread: Probability axioms problems

1. ## Probability axioms problems

Problem #1

If P(a) = .4 , P(b)= .5 and P(a ∩ b) = .3 find the following

a). P (a U b) = .6 because P (a U b) = P(a) + P(b) - P(a ∩ b)

b.) P(a' U b') = ?

c.) P(a U b') = .3 because P (a U b) = P(a ∩ b') + P(a ∩ b)

It's b.) that I'm not sure of since they're both compliments and I don't know what formula to use

Also Problem #2 (separate probabilities from #1)

If S = a U b , P(a) = .7, P(b) = .9, find

P(a ∩ b) = ?

and

P( (a ∩ b)' ) = ?

For these two problems I'm not really sure where to begin. Would S simply be 1 since P(S) is all elements of the sets and and b?

2. ## Re: Probability axioms problems

for (b)

$$\left(a \bigcap b \right) = \left(a' \bigcup b' \right)$$

for problem 2 S is just S. You don't know what it is so just call it S. Your answer will include it.

use the above formula. use (1-p) for the probability of the complement

for the 2nd one just do what you did in 1c with different numbers and the unknown S.

3. ## Re: Probability axioms problems

Thank you very much. As far as problem two is concerned, for the second part P( (a ∩ b)' ) is that the same as P(a' ∩ b') = P(a U b) ? because for P(a ∩ b) I got S. and for the 1st part of the problem I got S = .7 + .9 - P(a ∩ b) and thus P(a ∩ b)= 1.6 - S.

4. ## Re: Probability axioms problems

oh my mistake (c) was the union and you have the intersection here. Just apply demorgan's law

$$(a \bigcap b)'=a' \bigcup b'$$

and note that

$$S'=a'\bigcap b'$$

5. ## Re: Probability axioms problems

Hello, crownvicman!

Did you draw a Venn diagram?

$\displaystyle \text{If }P(A) = 0.4,\;P(B)= 0.5,\;P(A\cap B) = 0.3,\;\text{ find the following:}$

. . $\displaystyle (a)\;P(A \cup B) \qquad (b)\;P(A' \cup B') \qquad (c)\;P(A \cup B')$
Code:
    *-------------------------------------*
|                                     |
|  U        * * *       * * *         |
|       *   A       *       B   *     |
|     *           *   *           *   |
|    *           *     *           *  |
|                                     |
|   *           *       *           * |
|   *    0.1    *  0.3  *    0.2    * |
|   *           *       *           * |
|                                     |
|    *           *     *           *  |
|     *           *   *           *   |
|       *           *           *     |
|  0.4      * * *       * * *         |
|                                     |
*-------------------------------------*
Now you can answer the questions.

$\displaystyle \text{If }S \,=\,A\cup B,\; P(A) = 0.7,\;P(B) = 0.9,$

. . $\displaystyle \text{Find: }(1)\;P(A\cap B)\;\;\;(2)\;P(A\cap B)'$

I assume that $\displaystyle S$ is the universal set.

Then the diagram looks like this:

Code:
              * * *       * * *
*   A       *       B   *
*           *   *           *
*           *     *           *

*            *      *           *
*    0.1    *  0.6  *    0.3    *
*           *       *           *

*           *     *           *
*           *   *           *
*           *           *
* * *       * * *