We are given `P(A)=.4 ` , `P(bar(B))=.2 ` , and `P(A uu B)=.7 `

(a) Since the probability of the complement of B is .2, the probability of B is .8

(b) The probability of A and B is `P(A nn B)=P(A)P(B)=.4*.8=.32 `

(c) The probability that A occurs and B does not is `P(A nn bar(B))=.4*.2=.08 `

(d) The probability that A does not occur and that B does not occur is

`P(bar(A) nn bar(B))=(.6)(.2)=.12 `

(e) The probability that neither A nor B occurs is `P(bar(A uu B))=1-.7=.3 `

a) You're given the complement of B, so to find the probability of B, simply subtract the complement's probability from 1. So `P(B)=1-P(barB)=1-0.2=0.8`

b) The probability of A and B is the probability of the two individual events multiplied together, so `P(AnnB)=P(A)P(B)=0.4*0.8=0.32`

c) This probability is equal to the probability of A, and the probability of NOT B. So `P(AnnbarB)=P(A)P(barB)=0.4*0.2=0.08`

d) Neither A nor B occurs is the complement of A **or** B, and you're given this probability in the quesiton. So`P(bar(AuuB))=1-0.7=0.3`