Probability question Stat

Question

with events X, Y, and Z such that X and Y are mutually exclusive, X is a subset of Z, P(Y)= 1/2 ,P(Z)= 3/4, P(Y and Z)= 3/8, and P(X and Z)= 1/4..... what is P(X), P(Z and not X), P(Y or Z),P(not X or Y), P(Y|not Z)

Answer 1

• First of all, if X is a subset of Z, then P(X and Z) is the same as P(X).  So P(X) = 1/4.

• P(Z and not X) would be the part of Z NOT in X, or P(Z) - P(X) = 3/4 - 1/4 = 1/2.

• P(Y or Z) = P(Y) + P(Z) = P(Y and Z) = 1/2 + 3/4 - 3/8 = 7/8.

• P(not X or Y) = P(not X) + P (Y) - P ((not X) and Y).  But since X and Y are mutually exclusive, there is no overlap between X and Y, and Y is a subset of of not X.  So P((not X) and Y) is the same as P(Y).  So P(not X) + P (Y) - P ((not X) and Y) = P(not X) + P (Y) - P(Y) = P(not X) = 1 - P(X) = 1 - 1/4 = 3/4.

• Finally, by definition P(Y | not Z) = P (Y and not Z) / P (not Z).   To calculate this, remember two things:

1. P(Y) = P(Y and Z) + P(Y and not Z).  So P(Y and not Z) = P(Y) - P(Y and Z) = 1/2 - 3/8 = 1/8. 
2. P(not Z) = 1 - P(Z) = 1 - 3/4 = 1/4. 

 

So P (Y and not Z) / P (not Z) = 1/8 / 1/4 = 1/2.