Find P(A or B)

Question

Suppose that A and B are events and the following probabilities are known: P(A)=0.4, P(B)=0.3 and P(A given B)=0.2. Fine P(A or B)  

Answer 1

In general, 

 

P(A or B)= P(A)+P(B)-P(A and B)

 

Let's put this formula into perspective. Pretend I wanted to counted all the types of books on my bookshelf. If I count the Math textbooks, say there are 10, and the textbooks that are yellow, say there are 9, does that mean there are at least 19 books on my bookshelf? No! But why? Well, there might be Math textbooks which might also be yellow. If there are, then I have counted these books twice - once when I counted the Math books and once when I counted the yellow textbooks. I need to remove the double counts. How many double counted books are there? Well that is all the textbooks that are yellow and about Math. Say there are 3. Then I have at least 10+9-3 books on my bookshelf. That is why how the formula works. It naively adds everything together - P(A)+P(B) - then subtracts the things that might be double counted - P(A and B). This formula works equally well whether we are given the number of things or their probabilities - as we are in this problem.

 

Here, we are given P(A)=0.4 and P(B)=0.2 but instead we are given P(A|B)=P(A given B)=0.2. Well, 

 

P(A|B)=P(A given B)=P(A and B)/P(B).

 

But we want P(A and B). However, we do know P(A given B) and P(B). So we can solve for P(A and B):

 

P(A and B) = P(B) P(A given B)= 0.3*0.2=0.06

 

Therefore,

 

P(A or B)=P(A)+P(B)-P(A and B)=0.4+0.3-0.06=0.64

Answer 2

 Mohammed, P(A given B) = P(A and B)/P(B)

 

Plugging in what we have 0.2 = P(A and B)/0.3.

 

Multiplying both sides by 0.3, we get P(A and B) = 0.06

 

Now, P(A or B), since they are dependent, = P(A) + P(B) - P(A and B) = 0.4 + 0.3 - 0.06 = .64.

 

Hope this helps!