## Introduction to Probability Models

Lecture 5

Qi Wang, Department of Statistics

Aug 30, 2017

## Example 1

In his senior year, Brad decides to apply to medical school. He sends
his MCAT scores to two medical schools, Indiana and Johns Hopkins. Based on
how his friends fared in their med school applications, Brad estimates that
the probability of being accepted at Indiana is 0.7 and the probability of being
accepted at Johns Hopkins is 0.4. The probability of being rejected by both Indiana
and Johns Hopkins is 0.15.

Let J = accepted at Johns Hopkins; I = accepted at Indiana

- Are J and I mutually exclusive?
- Are J and I exhaustive? Do J and I form a partition of the sample space?
- Draw a labeled Venn diagram that describe the problem above.
- Find \(P(J\cup I) \)
- What is the probability that Brad was accepted to both Indiana and Johns Hopkins?
- What is the probability that Brad was accepted to Indiana only?
- Knowing that Brad was accepted to Indiana, what was the probability that he was also accepted to Johns Hopkins?

## Example 2

A fair six-sided die is rolled twice

- Write out the sample space using correct notation:
- Define the following events:
- J = the two rolls are the same number;
- K = the sum of the rolls is at least 4;
- L = the sum of the rolls is 7.

FIND \(P(J), P(K), P(L)\)
- What is \(P(K^c)\)
- Can two of those events happen on the same set of 2 rolls of the die?