Question

WILL GIVE MEDAL!!!!!!!!!

Events A_1, A_2 and A_3 form a partiton of the sample space S with probabilities P(A_1) = 0.1, P(A_2) = 0.1, P(A_3)= 0.8.
If E is an event in S with P(E|A_1) = 0.3, P(E|A_2) = 0.5, P(E|A_3) = 0.4, compute

P(E)=?
P(A1|E)=?
P(A2|E)=?
P(A3|E)=?

Answer 1

The symbol P(A|B) is called the conditional probablity and is defined as:
P(A|B) = P(A∩B)/P(B) which can also be written as:
equation 1: P(A∩B) = P(A|B)P(B)

Notice that since A1,A2, and A3 form a partition of S, this means that A1∩A2=A1∩A3=A2∩A3=∅ and A1∪A2∪A3=S.

Therefore,for any event E, E∩A1, E∩A2, and E∩A3 are disjoint and P(E)= P(E∩A1)+P(E∩A2)+P(E∩A3). Now by equation (1), we can re-write the above as:
P(E)= P(E|A1)P(A1) + P(E|A2)P(A2) + P(E|A3)P(A3)

Now we can compute P(E).
P(E)=(0.2)(0.3)+(0.3)(0.3)+(0.7)(0.4) = .06+.09+0.28 = 0.43

P(A1/E)= P(A1∩E)/P(E) = P(E/A1)P(A1)/P(E) = (0.2)(0.3)/(0.43) = 0.1395

P(A2/E)= P(A2∩E)/P(E) = P(E/A2)P(A2)/P(E) = (0.3)(0.3)/(0.43) = 0.2093

P(A3/E)= P(A3∩E)/P(E) = P(E/A3)P(A3)/P(E) = (0.7)(0.4)/(0.43) = 0.6511

For a good over-view of conditional probability see the link below.
http://mm.icu.ac.kr/menu2/popup/2009SICE...

Answer 2

did you get this from google/yahoo answers because I tried this and its not correct

Answer 3

the idea is correct...the numbers are wrong

Answer 4

yeah I know, would you happen to know the correct answer?

Answer 5

I do...use the formulas given above...but use the numbers given in your particular problem

Answer 6

I tried but I'm a little confused.. what answers did you get?

Answer 7

I don't give out answers