probability

Question

a professor of economics has 2 graduate assistants helping him. the probability that the older of two assistants will be absent on any given day is 0.08, the younger one 0.05, the probability that both will be absent is 0.02. find the probabilities that 1. either or both of them will be absent 2. at least one of them will not be absent 3. only one of them will be absent

Answer 1

Usually, we would calculate the probability that both are absent by multiplying the probabilities that either are absent.  This would give us .08*.05=.004.  However, in this case, the probability of both being absent is given as .02, different from the number we calculated, which tells us that they are not mutually exclusive events, so we will need to use the procedures for non-mutually exclusive sets.

 

1. Find the probabilities that either or both of them will be absent.

For this, we must add the probabilities of both events, and subtract the overlap since either of them being absent is included in both being absent.

.08+.05-.02=.11.

 

2. At least one of them will not be absent.

We can reword this as the probability of them not both being absent.  probability of both being absent is .02.  So,

1-.02=.98

 

3.  Only one of them will be absent.

We can reword this as the probability of not both being absent or both being present.

The probability of both being absent is given as .02

The probability of both being present is the complement of the answer in part 1.  1-.11=.89

This means the probability of not both being absent or both being present is the complement of these two events added.

1-(0.89+.02)

=1-.91

=.09

Answer 2

For this problem, you can create a Venn Diagram. One circle will represent the older assistant being absent and the other circle will represent the younger assistant being absent. Their overlap represents both assistants being absent, and the space outside the circles represents neither assistant being absent.

 

Let's fill some numbers in.

 

We'll start with the circles' overlap. You're told that the probability both assistants are absent is 0.02, so write that in the overlap area.

 

The older assistant has a 0.08 chance of being absent. Subtracting the 0.02 leaves you with 0.06 to put in the older-assistant-absent circle. Similarly, the younger assistant's 0.05 chance of absence means you should put 0.03 in the younger-assistant absent circle.

 

Total, that's 0.02 + 0.06 + 0.03 = 0.11 so far. So, you should have 1 - 0.11 = 0.89 outside both the circles. That way, the probabilities all add up to 1.

 

 

 

Now we can answer your questions. The probability that either or both of them will be absent is:

 

only younger assistant absent + only older assistant absent + both absent

0.03 + 0.06 + 0.02

0.11

 

The probability that at least one of them will not be absent is:

 

neither absent + only younger assistant absent + only older assistant absent

0.89 + 0.03 + 0.06

0.98

 

The probability that only one of them will be absent is:

 

only younger assistant absent + only older assistant absent

0.03 + 0.06

0.09