Math 121 - Week 1 Notes

Monday, September 7

This week we’ll start talking about probability theory. Today, we covered most of the material from Section 3.1 in the book.

We started by describing probability models which have two parts:

1. A set of possible outcomes called the sample space, and
2. A probability function which assigns a probability to subsets of the sample space (which are called events).

Even if the terminology is new, you are already familiar with probability models. We looked at several examples, such as a coin toss or rolling dice.

When you have a probability model, you can draw a probability histogram, also known as a probility distribution. We did this in the Probability Distributions Workshop.

Notice in the workshop problem 4(b) uses the notation \(P(X\ge15)\). This is shorthand for “The probability of X being at least 15”. In general, the symbol \(P( )\) will denote the probability of whatever is inside the parentheses.

We also talked about the rules of probability.

Rules of Probability

1. The probability of an event is always between 0 and 1.
2. The sum of the probabilities of all of the outcomes in a sample space is 1.
3. \(P(\text{An event doesn't happen}) = 1 - P(\text{The event happens})\) (Complimentary events rule)
4. \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\) (Addition Rule)
5. If \(A\) and \(B\) are independent events, then \(P(A \text{ and } B) = P(A) \cdot P(B)\) (Multiplication rule for independent events)

Two events \(A\) and \(B\) are independent if knowing that one happens has no affect on the probability that the other happens.

We finished by drawing a Venn diagram (like the one below) to illustrate this practice problem:

1. Suppose that you need to have knee surgery. The surgeon explains the risks to you. There is a 11% chance that the surgery will not fix your knee problem. There is a 4% chance you will get an infection. And there is a 2% chance that both will happen. What is the probability that the surgery succeeds without an infection?

Wednesday, September 9

Today we introduced tree diagrams as a way to keep the multiplication rule straight. We talked about how to draw a weighted tree diagram, and how to use it to solve probability problems. We did several examples, including this Tree Diagrams workshop.