Let E and F be two mutually exclusive events, and suppose that P(E) = .2 and P(F) = .5

Question

Let E and F be two mutually exclusive events, and suppose that P(E) = .2 and P(F) = .5.
Compute the following probabilities:

 

a. P(E ∩ F)

b. P(E ∪ F)

 

c. P(E^c)

 

d. P(E^c ∩ F^c)

 

e. P(E^c ∪ F^c)

 

 

Answer 1

a)  Since the events are mutually exclusive, they cannot both occur simultaneously.  The probability is 0.

 

b) We can use the formula P(A u B)=P(A) + P(B) - P(A n B).  Notice that P(A n B) = 0, from part a.  This leaves

P(A u B) = P(A) + P(B) = .2 + .5 = .7

 

c)  The probability of the complement of an event E is 1 - P(E).

Here, we have P(E^c) = 1 - P(E) = 1 - .2 = .8

 

For the next two problems, a Venn diagram might be useful. Since the events are mutually exclusive, they can be drawn as two non-overlapping circles. 

 

d)  The shaded region of the corresponding Venn diagram excludes only the interiors of E and F.  That is, the probability is equal to 1 - P(E u F) = 1 - .7 = 0.3.

 

e)  E^c contains all of F, and F^c contains all of E.  When taking the union of the shaded regions corresponding to E^c and F^c, the entire diagram is shaded in.  So, the probability is 1.