Answer:
a) ![P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.3}{0.45}= 0.667](https://tex.z-dn.net/?f=%20P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D%20%3D%20%5Cfrac%7B0.3%7D%7B0.45%7D%3D%200.667)
Represent the probability that the event B occurs given that the event A occurs first
b) ![P(B'|A) = \frac{0.15}{0.45}=0.333](https://tex.z-dn.net/?f=%20P%28B%27%7CA%29%20%3D%20%5Cfrac%7B0.15%7D%7B0.45%7D%3D0.333)
Represent the probability that the event B no occurs given that the event A occurs first
c) ![P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.3}{0.35}= 0.857](https://tex.z-dn.net/?f=%20P%28A%7CB%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28B%29%7D%20%3D%20%5Cfrac%7B0.3%7D%7B0.35%7D%3D%200.857)
Represent the probability that the event A occurs given that the event B occurs first
d) ![P(A'|B) = \frac{0.05}{0.35}=0.143](https://tex.z-dn.net/?f=%20P%28A%27%7CB%29%20%3D%20%5Cfrac%7B0.05%7D%7B0.35%7D%3D0.143)
Represent the probability that the event A no occurs given that the event B occurs first
Step-by-step explanation:
For this case we have the following probabilities given for the events defined A and B
![P(A) = 0.45, P(B) = 0.35, P(A \cap B) =0.30](https://tex.z-dn.net/?f=%20P%28A%29%20%3D%200.45%2C%20P%28B%29%20%3D%200.35%2C%20P%28A%20%5Ccap%20B%29%20%3D0.30)
For this case we can begin finding the probability for the complements:
![P(B') =1-P(B) = 1-0.35= 0.65](https://tex.z-dn.net/?f=%20P%28B%27%29%20%3D1-P%28B%29%20%3D%201-0.35%3D%200.65)
![P(A') =1-P(A) = 1-0.45= 0.55](https://tex.z-dn.net/?f=%20P%28A%27%29%20%3D1-P%28A%29%20%3D%201-0.45%3D%200.55)
For this case we are interested on the following probabilities:
Part a
![P(B|A)](https://tex.z-dn.net/?f=%20P%28B%7CA%29)
For this case we can use the Bayes theorem and we can find this probability like this:
![P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.3}{0.45}= 0.667](https://tex.z-dn.net/?f=%20P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D%20%3D%20%5Cfrac%7B0.3%7D%7B0.45%7D%3D%200.667)
Represent the probability that the event B occurs given that the event A occurs first
Part b
![P(B'|A) = \frac{P(B' \cap A)}{P(A}](https://tex.z-dn.net/?f=%20P%28B%27%7CA%29%20%3D%20%5Cfrac%7BP%28B%27%20%5Ccap%20A%29%7D%7BP%28A%7D)
And for this case we can find ![P(B' \cap A) =P(A) -P(A\cap B)= 0.45-0.3=0.15](https://tex.z-dn.net/?f=%20P%28B%27%20%5Ccap%20A%29%20%3DP%28A%29%20-P%28A%5Ccap%20B%29%3D%200.45-0.3%3D0.15)
And if we replace we got:
![P(B'|A) = \frac{0.15}{0.45}=0.333](https://tex.z-dn.net/?f=%20P%28B%27%7CA%29%20%3D%20%5Cfrac%7B0.15%7D%7B0.45%7D%3D0.333)
Represent the probability that the event B no occurs given that the event A occurs first
Part c
![P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.3}{0.35}= 0.857](https://tex.z-dn.net/?f=%20P%28A%7CB%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28B%29%7D%20%3D%20%5Cfrac%7B0.3%7D%7B0.35%7D%3D%200.857)
Represent the probability that the event A occurs given that the event B occurs first
Part d
![P(A'|B) = \frac{P(A' \cap B)}{P(B}](https://tex.z-dn.net/?f=%20P%28A%27%7CB%29%20%3D%20%5Cfrac%7BP%28A%27%20%5Ccap%20B%29%7D%7BP%28B%7D)
And for this case we can find ![P(A' \cap B) =P(B) -P(A\cap B)= 0.35-0.3=0.05](https://tex.z-dn.net/?f=%20P%28A%27%20%5Ccap%20B%29%20%3DP%28B%29%20-P%28A%5Ccap%20B%29%3D%200.35-0.3%3D0.05)
And if we replace we got:
![P(A'|B) = \frac{0.05}{0.35}=0.143](https://tex.z-dn.net/?f=%20P%28A%27%7CB%29%20%3D%20%5Cfrac%7B0.05%7D%7B0.35%7D%3D0.143)
Represent the probability that the event A no occurs given that the event B occurs first