Question

In a certain population, the probability that a woman has red hair is 1/5. If a woman has red hair, the probability of her having freckles is 6/7. If she does not have red hair, the probability of her having freckles is 1/25. If a woman is chosen at random from the population, what is the probability that she will have freckles? I know the answer is supposed to be 178/875 but how?

Answer 1

Do you know how to make a conditional probabilities tree diagram?

Answer 2

Those can help keep all the numbers straight.

Answer 3

no I do not know about a conditional probability diagram

Answer 4

Alright, I'll draw one up for you.

(In the meantime, for general reference, see http://stattrek.com/probability/probability-rules.aspx?Tutorial=Stat )

Answer 5

W = Set of all women
R = (R | W) = probability of red hair (given woman)
-R = (R | W) = probability of not having red hair (given woman)
F | R = probability of having freckles given red hair
-F | R = probability of not having freckles given red hair
F | -R = probability of having freckles given no red hair
-F | -R = probability of not having freckles given no red hair

Answer 6

Does it make sense how I drew that?

Answer 7

What the question is asking for is P(F | W) = probability of having freckles given being a woman. This will equal the sum of P(F | R) + P(F | -R).

With me so far @amyg ?

Answer 8

yes, but I am having a hard time figguring what the probability is. I am confused about the 1/5 and 6/7 numbers. Can you plug them into the equasion and show me?

Answer 9

Sure. This is a combination of the addition rule (when you have an OR situation) and the multiplication rule (when you have an AND situation).

There are two ways to have a woman with freckles here: With red hair OR without red hair, so you are going to add those two conditional probabilities together.

Each of those conditionals is an AND situation.
a) having red hair AND having freckles
b) not having red hair AND having freckles

So you multiply those.

See if you can figure out how to put the numbers together from that.

Answer 10

So, if I add 6/7 and 1/25 I get 157/175. but multiplied they are 6/175. I'm lost, because I know the answer is supposed to be 178/875.

Answer 11

Multiply 1/5 and 6/7 to get Red Hair AND Freckles
Multiply 4/5 and 1/25 to get Not Red Hair AND Freckles
then add those two to get total probability of Freckles.

Answer 12

Here is one way to think of this

the red are 1/5 of the total
if we look at just that little 1/5 box, and divide it into 1/7 and 6/7

now in terms of the big box, that 1/7 is 1/5 of 1/7 = 1/35 (no freckles)
and the 6/7 is 1/5 of 6/7 or 6/35

now do the same for the non-reds.