An experiment results in one of three mutually exclusive events A, B, and C. It is known that P(A)= 0.30, P(B) = 0.55, and P(C) = 0.15. Find the following probabilities: a. P(A∪B)
b. P(A∩C)
c. P(A/B)
d. P(B∪C)

a. P(A∪B)=0.85

b. P(A∩C)=0

c. P(A/B)=0

d. P(B∪C)= 0.70

Step-by-step explanation:

Events are said to be mutually exclusive when both cannot occur simultaneously in the result of experimentation. They are also known as incompatible events.

Being P(A)=0.30,  P(B)= 0.55 and P(C)= 0.15

Let A and B be any two events, P (A) and P (B) the probability of occurrence of events A or B, is known as the probability of union [denoted as P (A U B)]:

P(A∪B)=P(A) + P(B) - P(A∩B)

Mutually exclusive events are results of an event that cannot occur at the same time. So:

P(A∩B)=0 That is, there is no chance that both events will occur.

So: P(A∪B)=P(A) + P(B)

In this case: P(A∪B)=P(A) + P(B)= 0.30 + 0.55 → P(A∪B)=0.85

As mentioned, if two events are mutually exclusive, there is no chance that both events will occur. Being the intersection an operation whose result is made up of the non-repeated and common events of two or more sets, that is, given two events A and B, their intersection is made up of the elementary events that they have in common, then: P(A∩C)=0

The probability that event A will occur if event B has occurred is called the conditional probability and is defined:

P(A/B)=P(A∩B)÷P(B) being P(B)≠0

Since A and B are mutually exclusive, P (A∩B) = 0. So:

P(A/B)=P(A∩B)÷P(B)=0÷0.55 → P(A/B)=0

Finally, P(B∪C)=P(B) + P(C) - P(B∩C)

Since A and B are mutually exclusive, P (B∩C) = 0. So:

P(B∪C)=P(B) + P(C)= 0.55 + 0.15 → P(B∪C)= 0.70

a. the residual plot for model 1 has a random pattern and is a good fit for the data.