An experiment can result in one of five equally likely simple events, E1, E2, , E5. Events A, B, and C are defined as follows. A: E2, E4 P(A) = 0.4 B: E1, E3, E4, E5 P(B) = 0.8 C: E2, E3 P(C) = 0.4 Find the probabilities associated with the following events by listing the simple events in each.

(1)

The notation means "the complement of event A".

Therefore, here we want to find , which is the probability for event A NOT to occur.

Here we are given that the probability of event A is

The probability of the complement of an event X is given by

So in this case,

(2)

Event A is defined as either event E2 or event E4 to occur.

Event B is defined as either one of the following events to occur: E1, E3, E4, E5.

Here we want to find the intersection between the sets of events A and B.

We see that this intersection consists of the events that belong simultaneously to both sets: therefore, only E4.

Therefore,

And since the 5 events are equally likely to occur, the probability of each is 0.2, therefore

(3)

Event B is defined as either one of the following events to occur: E1, E3, E4, E5.

Event C is defined as either E2 or E3 to occur.

Here we want to find the intersection between the sets of events B and C.

This intersection consists of the events that belong simultaneously to both sets, so, only E3.

Therefore,

And since each event E has probability 0.2 to occur,

(4)

Here we want to find the probability of the set consisting of the union of A and B.

We have:

Event A is defined as either event E2 or event E4 to occur.

Event B is defined as either one of the following events to occur: E1, E3, E4, E5.

So the union of the two sets is: E1, E2, E3, E4, E5

But these events represents all the possible events of the experiment. Therefore, their total probability is 1, so

(5)

Here we want to find the conditional probability of B given C: that is, the probability of B to occur, given that C has occurred.

This probability can be calculated as:

Here in this problem, we have:

(already calculated in part 3)

(given by the problem)

Therefore, we have:

(6)

Here we want to find the conditional probability of A given B: that is, the probability of A to occur, given that B has occurred.

This probability can be calculated as:

In this part of the problem, we have:

(already calculated in part 2)

(given by the problem)

And so, we have:

(7)

Here we want to find the probability that either A, either B or either C occurs.

Event A is defined as either event E2 or event E4 to occur.

Event B is defined as either one of the following events to occur: E1, E3, E4, E5.

Event C is defined as either E2 or E3 to occur.

We see that the 3 events A, B and C contain all the possible events of the experiment: E1, E2, E3, E4 or E5. Therefore, if any of these 5 events occurs, then either A, B or C has occurred as well.

Therefore, the total probability is 1:

(8)

Here we want to find the probability of the complement of the set

The probability of this set was already calculated in part 2, and it was

We also said that the probability of the complement of a set X is given by

Therefore, the probability of the complement of is:

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