SOLUTION: For two events A and B, P(A) = 0.3 and P(B)=0.2. (a) If A and B are independent, then P(A|B) = P(A∩B) = P(A∪B) = (b) If A and B are dependent and P(A|B) = 0.25, then P(

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Question 1180354: For two events A and B, P(A) = 0.3 and P(B)=0.2. (a) If A and B are independent, then P(A|B) =
P(A∩B) =
P(A∪B) =
(b) If A and B are dependent and P(A|B) = 0.25,
then P(B|A) =
P(A∩B) =
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52797)   (Show Source): You can put this solution on YOUR website!
For two events A and B, P(A) = 0.3 and P(B)=0.2. (a) If A and B are independent, then P(A|B) =
P(A∩B) =
P(A∪B) =
(b) If A and B are dependent and P(A|B) = 0.25,
then P(B|A) =
P(A∩B) =
~~~~~~~~~~~~~~~~~~~


                    I will solve part  a),  ONLY.


I will answer in other, more natural order. 


First, P(A ∩ B) = P(A)*P(B) = 0.3*0.2 = 0.06  (since A and B are independent).


Second,  P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.2 - 0.06 = 0.44.


Third, P(A | B) =  =  = 0.3.

Solved.


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Edwin expressed  "a favor"  in my address - - - so,  I will express my favor in his address,  mutually.

Ignore his solution:  it is  INCORRECT:

            It is incorrect, because it is based on your incorrect formulation of the part  b).

            The event  A  and  B  must be  INDEPENDENT - - - while your formulation
            calls them  "DEPENDENT"  - - -  clearly,  directly,  explicitly and incorrectly.


            And  Edwin  NEITHER  disproved  NOR  corrected it . . .



Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Pay no attention to her gripes.

For two events A and B, P(A) = 0.3 and P(B)=0.2. (a) If A and B are independent, then
P(A|B) = P(A∩B) =
P(A∪B) =
Venn diagrams make things easier to understand than formulas alone.

Draw a Venn diagram with sets A and B, with r,s,t,u representing the
probabilities of the four regions:



P(A) = 0.3 = r+s
P(B) = 0.2 = s+t

Since they are independent,

P(A∩B) = s = P(A)P(B) = (0.3)(0.2) = 0.06    <--ANSWER to P(A∩B)

So since

s = 0.06 and
r+s = 0.3, substitution gives

r+0.06 = 0.3
r = 0.3-0.06
r = 0.24

And since

s = 0.06 and
s+t = 0.2
0.06+t = 0.2
t = 0.14

P(A∪B) = r+s+t = 0.24+0.06+0.14 = 0.44   <--ANSWER to P(A∪B)



Note: We weren't asked for P(A'∩B') = u, but we could have
found it because r+s+t+u = 1, so 

 0.24+0.06+0.14+u = 1
           0.44+u = 1
                u = 0.56 

------------------------------------

(b) If A and B are dependent and P(A|B) = 0.25,
then P(B|A) =
P(A∩B) =



P(A|B) = P(A∩B)/P(B) = s/0.2 = 0.25
                           s = 0.25(0.2)
                           s = 0.05

P(B|A) = P(B∩A)/P(A) = P(A∩B)/P(A) = s/0.2 = 0.05/0.3 = 5/30 = 1/6
                     
(We could have just divided 0.05 by 0.3 and gotten 0.1666666...,
but repeating decimals are a hassle, so I multiplied top and bottom
by 100 and got the fraction 5/30 which reduced to 1/6.

We aren't asked to, but we could finish out the Venn diagram for (b) if we
like:




Edwin
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