# Simple probability question; why the calculation is as it is

Well the task is as follows.
About two events A and B, we know that $P(A)=\frac{1}{6}$ and $P(B)=\frac{2}{7}$. Find $P(A \cup B)$
a) If A and B is disjunktive
b) If A and B is independent

My solutions
a) $\frac{1}{6}+\frac{2}{7}$
b) $\frac{1}{6}+\frac{2}{7} - (\frac{1}{6} \times \frac{2}{7})$
This gives the correct answer but I'm not 110% sure why.

And, the same problem with another task.

For a girl to be colorblind both her parents needs to carry the gene for it. She inherits the gene from both parents independently. The probability is 8% that the girl gets that gene from the mother and 8% for she to get it from the father.
What is the probability that one girl is colorblind?
What is the probability that one girl is not colorblind?

The first one is just: $0.08^2$. That's fine.
The second one is $0.92+0.92 - 0.92^2$

I don't understand why the second isn't just $0.92^2$ like the first one.

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(+1) because I want to see more of such users and questions on this site. –  The Chaz 2.0 Jul 15 '11 at 20:48
Thanks for the upvote. I need to understand this 110%, I need an A in this class. –  Algific Jul 15 '11 at 21:15
Just remember that understanding, like probability, can never exceed 100%!  ;) –  The Chaz 2.0 Jul 15 '11 at 21:41
I wish I could give more than one upvote for that comment. –  mathmath8128 Jul 15 '11 at 21:45
For b), use that if A,B are independent, then P(A|B)=P(A)=$\frac {P(A\cap B)}{P(B)}$ , so that $P(A)P(B)=P(A\cap B)$ , then $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ –  gary Jul 16 '11 at 0:28

## 1 Answer

The formula is the same for both. The probability of their union is the sum of their (individual) probabilities, minus the probability of their intersection.

In a) their intersection is 0.

For the second problem, we have the exact same principle at work: add their individual probabilities (0.92 + 0.92), and subtract the probability that she inherited the gene from both parents (i.e. 0.92^2).

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Seeing when the intersection is empty is my problem. Intuitively you can still multiply them together and get a would-be intersection even if it would be empty. –  Algific Jul 15 '11 at 21:10
Sorry... How would that be intuitive, exactly? Intuition should tell you if/when events are exclusive, in the case that you aren't given that information explicitly. –  The Chaz 2.0 Jul 15 '11 at 21:40
Not intuitive. But mindlessly plugging in numbers in the wrong formula during the exam could happen. I understand it now, there are three outcomes where she is cblind and one where she isn't. Thanks for the help. –  Algific Jul 16 '11 at 14:09