so i am asked to find the prob of a male professor - given that he is promoted.
and the stats are
female promoted = 3%
female not promoted - 12%
male promoted 17%
male not promoted - 68%
and i just wrote down that it is 17% is this true?
ALso, i looked up in my book that P(A I B) = P(B and A)/P(A)
but my question is isnt P(b and a) = P(b) * P(a) ? so dosnt that cancel out with the denominator?
thank you!
You are given the probability of being male and being promoted = .17 = P(b and a)
To find the probability of a male professor being promoted, given that he is promoted, you need to use conditional probability.
First, let's define the events:
A: Male professor is promoted.
B: Professor is promoted.
The probability of a male professor being promoted, given that he is promoted, can be computed using the formula for conditional probability:
P(A | B) = P(A and B) / P(B)
Now, let's substitute the probabilities given in the problem into the formula:
P(A | B) = (male promoted) / (promoted)
According to the provided statistics, the probability of a male professor being promoted is 17%, and the probability of any professor being promoted (regardless of gender) is the sum of the probabilities of being promoted for both genders, which is (female promoted + male promoted) = 3% + 17% = 20%.
Therefore, the probability of a male professor being promoted, given that he is promoted, is:
P(A | B) = 17% / 20% = 0.17 / 0.20 = 0.85 or 85%.
Regarding your second question, P(A and B) denotes the probability of both events A and B occurring simultaneously. It is not the product of P(A) and P(B) unless A and B are independent events. Therefore, in this case, the formula P(A | B) = P(A and B) / P(B) cannot be simplified to P(A | B) = P(B) * P(A) / P(A).
I hope this clarifies your doubts!