Examples

Lucia is Host or Player

The professor chooses three students at random for a game in a class of 40, one to be Host, one to be Player, one to be Judge. What is the probability that Lucia is either Host or Player?

Solution

(1) Set up the probability model:

Label the students to . Write for Lucia’s number.

Outcomes: assignments such as . These are ordered triples with distinct entries in .

Sample space: is the collection of all such distinct triples

Events: any subset of

Probability measure: assume all outcomes are equally likely, so for all

In total there are triples of distinct numbers.

Therefore for any specific outcome .

Therefore for any event . (Recall is the number of outcomes in .)

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(2) Define the desired event:

We want to find . Define and . Thus:

So, in this notation, we seek .

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(3) Compute the desired probability:

Importantly, (mutually exclusive). There are no outcomes in in which Lucia is both Host and Player.

By additivity, we infer .

Now compute . There are ways to choose and from the students besides Lucia. Therefore . Therefore:

Now compute . It is similar: .

Finally compute that , so the answer is:

iPhones and iPads

At Mr. Jefferson’s University, 25% of students have an iPhone, 30% have an iPad, and 60% have neither.

What is the probability that a randomly chosen student has some iProduct? (Q1)

What about both? (Q2)

Solution

(1) Set up the probability model:

A student is chosen at random:

Outcomes are chosen students.

The sample space is the set of all students. Events are subsets of .

Write and (regarding the chosen student).

All students are equally likely to be chosen. Therefore for any event . Therefore and .

Furthermore, . This states that 60% have “not iPhone AND not iPad”.

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(2) Define the desired event:

Q1:

Q2:

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(3) Compute the probabilities:

We do not know that and are exclusive.

We could try inclusion-exclusion:

We know and . So this formula, with given data, RELATES Q1 and Q2. It does not solve either one by itself.

We have not yet used the information that .

To use this, simplify it with De Morgan’s Laws:

Therefore:

We have answered Q1. Recall that inclusion-exclusion relates Q1 and Q2 and solve to answer Q2: