SOLUTION: P(A) = 0.5 P(B) = 0.8 P(A ᑎ B ) = 0.3 Solve this question 1. P ( A|A ᑎ B) 2. P ( A ᑎ B | A ᑌ B )

Algebra ->  Probability-and-statistics -> SOLUTION: P(A) = 0.5 P(B) = 0.8 P(A ᑎ B ) = 0.3 Solve this question 1. P ( A|A ᑎ B) 2. P ( A ᑎ B | A ᑌ B )      Log On


   

Question 1160040: P(A) = 0.5
P(B) = 0.8
P(A ᑎ B ) = 0.3
Solve this question
1. P ( A|A ᑎ B)
2. P ( A ᑎ B | A ᑌ B )
Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
P( A ) = 0.5
P( B ) = 0.8
P( A ᑎ B ) = 0.3
Solve this question
1. P ( A | A ᑎ B )
2. P ( A ᑎ B | A ᑌ B )
The formula for conditional probability is

             P( X ᑎ Y )              
P( X | Y ) = ————————————
               P( Y )


1. P ( A | A ᑎ B )

Substitute A for X and A ᑎ B for Y

                  P[ A ᑎ (A ᑎ B) ]    P[ (A ᑎ A) ᑎ B ]    P( A ᑎ B )      
P ( A | A ᑎ B ) = ————————————————— = ————————————————— =  ——————————— = 1
                     P( A ᑎ B )          P( A ᑎ B )        P( A ᑎ B )

It is certain that you are given A if you are given A and B.
It doesn't even matter what the probabilities for A, B are.

2. P ( A ᑎ B | A ᑌ B )

Substitute A ᑎ B for X and A ᑌ B for Y

                      P[(A ᑎ B) ᑎ (A ᑌ B) ]              
P( A ᑎ B | A ᑌ B ) = ——————————————————————
                           P( A ᑌ B )

We simplify 

(A ᑎ B) ᑎ (A ᑌ B) = [(A ᑎ B) ᑎ A] ᑌ [(A ᑎ B) ᑎ B] =

[A ᑎ (B ᑎ A)] ᑌ [A ᑎ (B ᑎ B)] = [A ᑎ (A ᑎ B)] ᑌ (A ᑎ B) =

[(A ᑎ A) ᑎ B)] ᑌ (A ᑎ B) = (A ᑎ B) ᑌ (A ᑎ B) = A ᑎ B

The formula for

P( X ᑌ Y ) = P( X ) + P( Y ) - P( X ᑎ Y ) 

P( A ᑌ B ) = P( A ) + P( B ) - P( A ᑎ B )

P( A ᑌ B ) = 0.5 + 0.8 - 0.3 = 1

                       P( A ᑎ B )     0.3            
P( A ᑎ B | A ᑌ B ) = ————————————— = ———— = 0.3
                       P( A ᑌ B )      1     


Edwin