Total probability with different probabilities

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I have a silly question that has actually been kinda racking my brain since I have almost no formal education when it comes to probabilities. Let's say I know that a girl has a 40% chance of earning a college degree. Let's also say that right handed people have a 30% chance of earning a college degree. Then let's keep tossing in things like california girls have a 50% chance of getting a college degree. Obviously you can keep finding characteristics of that girl and find the statistics behind their success in getting a college degree.

Now the grand question is... what is the probability that a californian, right-handed, girl has of getting a college degree? I assume we can't simply multiply probabilities because it seems like you can keep throwing probabilities out like that and eventually make the probability nearly 0 even though it can't be...

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A = person goes to college, B = person is female, C = person is right-handed
You are given P(A|B) and P(A|C) and wish to find P(A|B,C). Let's assume that B,C are independent, and also assume that B,C are conditionally independent given A--that is, P(B,C|A) = P(B|A)P(C|A), and P(B,C) = P(B)P(C).
So,
P(A|B,C)=P(A,B,C)/P(B,C) = P(A)P(B,C|A)/P(B,C) = P(B|A)P(C|A) P(A)/(P(B)P(C)) = P(A|B)P(B)/P(A) P(A|C)P(C)/P(A) P(A)/(P(B)P(C)) = P(A|B)P(A|C)/P(A)

So instead of just multiplying by P(A|C), you multiply by P(A|C)/P(A)

CRGreathouse
Homework Helper
So instead of just multiplying by P(A|C), you multiply by P(A|C)/P(A)
Or, cutting out all the Baysian stuff (good, but if you haven't had a probability class I don't imagine it would make sense):

If a foo has a 30% chance of getting a degree, but a person (including foos and non-foos) has a 35% of getting a degree, you should be multiplying by .30/.35 rather than .30.

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Yah this is what sucks about never having taken a statistics class :(.

So let me make the problem more accurate and tell me if i do this right. A girl has a 40% chance of having a bachelors degree. Girls who are right handed have a 50% chance of having a degree. Girls who live in california have a 30% chance of having a degree.

Thus the probability that a girl who is right handed who lives in california will have a (.4*.4*.4)/(.3*.5) chance of having a degree? Thus a 42.6% chance?

What if say, the 2nd condition "girls who are right handed have a 50% chance of having a degree" was changed to "people who are right handed have a 50% chance of having a degree"?