Conditional Probability

[JellyFish]

I have a question involving conditional probabilty.

A group of 100 girls contains 30 blondes and 70 brunettes. Twenty-five of thevblondes are blue-eyed and the rest are brown-eyed, whereas 55 of the brunettes are brown-eyed and the rest are blue-eyed. A girl is chosen at random and it is determined that she is blue-eyed while her hair color was not visible because of her hat. Find the conditional probability that she is a blonde.

I thought I should solve this by letting A be the event that a girl has blue eyes, B be the event they have brown eyes and C be the event that they are blonde.
And then I would solve for c with p(C) = p(C|A)p(A) + p(C|A[sup:1tq2sroy]c[/sup:1tq2sroy])p(A[sup:1tq2sroy]c[/sup:1tq2sroy]) = p(C|A)p(A) + p(C|B)p(B). But doing this I'm not using anything to do with brunettes?

Also do I leave the numbers the way they are given or make them into percentages?

Thank you

 

[galactus]

Lets build a chart. That makes answering these easier.

\(\displaystyle \begin{tabular}{c|c|c|c} \;\ & \text{blond}&\text{brunette}&\text{total}\\\hline \text{blue}&25&15&40\\ \hline\text{brown}&5&55&60\\ \hline\text{total}&30&70&100\end{tabular}\)


We want the probability she is blond given she has blue eyes.

\(\displaystyle P(\text{blond}|\text{blue eyes})\)

Go down the blonde column to the blue row and go across to the total. 25/40=5/8

\(\displaystyle P(\text{Blond}|\text{Blue})=\frac{P(\text{Blond}\cap\text{Blue})}{P(\text{Blond}\cap\text{Blue})+P(\text{Brunette}\cap\text{Blue})}=\frac{\frac{25}{100}}{\frac{25}{100}+\frac{15}{100}}\)

 

[JellyFish]

Thanks Galactus, the chart made things much easier to see.

I am still a little confused about the formula you used though. Is this a "general" formula or something you've rearranged. I think the P(blueblue) is confusing me as well.
Sorry

 

[galactus]

That was a typo. Sorry. I fixed it. Should have been 'blond'.

The formula is kind of like Baye's theorem

 

[JellyFish]

I did find that version of Bayes' Formula in my text and when I carried it out I too got the probability of the girl being a blonde is 5/8. Thank you again.