| The probability of B occuring given that A has already occured can be calculated using :
P(B|A) = P(ANB) / P(A)
If B depend on A, then A depends on B and can be calculated as follows :
The probability of A occuring given that B has already occured can be calculated using :
(note P(A1) is the probability of A not happening and P(B|A1) is the probability of B if A doesn't happen)
P(A|B) = P(ANB) / P(B) = P(B|A) * P(A) / (P(B|A) * P(A) + P(B|A1) * P(A1)) |
The Staff at a school consists of three categories ; teachers, admin and maintenance
Both men and women work in these categories as follows;
men : teachers 20, admin 10 and maintenance 10
women : teachers 30, admin 10 and maintenance 20
Each month every member of staff is entered in a draw for a cash bonus and one person is selected at random to receive the bonus
Let A denote the event that the person selected is female.
Let B denote that the person selected is a teacher.
1. Evaluate P(A)
solution - count all the women and divide by the total number of people, giving the prob that the selected person is a woman P(A)
answer - P(A) = 30+10+20 /20+10+20+20+10+10 = 60/100 = 0.6
2. Evaluate P(B|A)
solution - we need to find the prob of B (the person selected is a teacher) given A (that the person is already known to be a woman)
so we need to divide the number of women teachers by the total number of women.
answer - P(B|A) = 30/60 = 0.5
3. Evaluate P(AUB)
solution : tricky we don't have enough info to solve P(AUB) = P(A) + P(B) - P(ANB) - we need to first find P(B) and P(ANB)
P(B)
is easy - similar to P(A) above but for teachers rather than women
- P(B) = count of teachers divided by total number of people
P(B) = 30+20/ 100 = 0.5
to find P(ANB) from the info we already have we must use P(B|A) = P(ANB) / P(A)
so P(ANB) = P(B|A) * P(A) = 0.5 * 0.6 = 0.3
answer - P(AUB) = 0.6 + 0.5 - 0.3 = 0.8
4. Are A and B independent?
The answer to this question is fairly easy -
If P(ANB) = P(A) * P(B) then they are independent
P(ANB) = 0.3 and P(A) * P(B) = 0.6 * 0.5 = 0.3
so P(ANB) = P(A).P(B) so A and B are independent.
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| notes
"Probability" is a big word - so I shorten it to "Prob"
that should not be a problem!
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