# Probability axioms problems

• Feb 17th 2014, 07:01 PM
crownvicman
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?
• Feb 17th 2014, 08:45 PM
romsek
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.
• Feb 17th 2014, 09:05 PM
crownvicman
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.
• Feb 17th 2014, 09:46 PM
romsek
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

$$S'=a'\bigcap b'$$
• Feb 18th 2014, 07:11 AM
Soroban
Re: Probability axioms problems
Hello, crownvicman!

Did you draw a Venn diagram?

Quote:

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

. . $\displaystyle (a)\;P(A \cup B) \qquad (b)\;P(A' \cup B') \qquad (c)\;P(A \cup B')$

Code:

    *-------------------------------------*     |                                    |     |  U        * * *      * * *        |     |      *  A      *      B  *    |     |    *          *  *          *  |     |    *          *    *          *  |     |                                    |     |  *          *      *          * |     |  *    0.1    *  0.3  *    0.2    * |     |  *          *      *          * |     |                                    |     |    *          *    *          *  |     |    *          *  *          *  |     |      *          *          *    |     |  0.4      * * *      * * *        |     |                                    |     *-------------------------------------*
Now you can answer the questions.

Quote:

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

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

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

Then the diagram looks like this:

Code:

              * * *      * * *           *  A      *      B  *         *          *  *          *       *          *    *          *       *            *      *          *       *    0.1    *  0.6  *    0.3    *       *          *      *          *       *          *    *          *         *          *  *          *           *          *          *               * * *      * * *