SOLUTION: Determine all joint probabilities listed below from the following information: P(A)=0.74,P(Ac)=0.26,P(B|A)=0.43,P(B|Ac)=0.65 P(A and B) = P(A and Bc) = P(Ac and B) = P(

Below is the final solution.

Edwin AKA AnlytcPhil

let w, x, y, and z represent the probabilities of the regions
they are in in the Venn diagram below



P(A and B) = "what's in both circles" = x
P(A and Cc) = "what's in A but not in B" = w,
P(Ac and B) = "what's not in A but is in B" = y,
P(Ac and Bc) = "what's not in A and not in B" = z



Simplifying the third and fourth equations:

    

We have this system:



Solve the system:



and get w = 0.4218, x = 0.3182

Solve the system:



and get y = 0.169, x = 0.091

Put those values in the Venn diagram:



Edwin

(1)  To calculate P(A and B), it is enough to have these two given values  P(A)= 0.74  and  P(B | A)= 0.43.


     By the definition of the conditional probability,  P(B | A) = P(A and B)/P(A).


     Therefore, to get P(A and B), simply multiply P(A | B) and P(A) to get


                P(A and B) = 0.74*0.43 = 0.3182.     ANSWER




(2)  P(A and Bc) = P(A) - P(A and B) = 0.74 - 0.3182 = 0.4218.    ANSWER




(3)  The solution for (3) is absolutely identical to the solution for (1).

     Therefore, I will repeat it word-in-word.



     To calculate P(Ac and B), it is enough to have these two given values  P(Ac)= 0.26  and  P(B | Ac)= 0.65.


     By the definition of the conditional probability,  P(B | Ac) = P(B and Ac)/P(Ac).


     Therefore, to get P(Ac and B), simply multiply P(B | Ac) and P(Ac) to get


                P(Ac and B) = 0.65*0.26 = 0.169.     ANSWER




(4)  P(Ac and Bc) = P(Ac) - P(Ac and B) = 0.26 - 0.169 = 0.091.    ANSWER