This material is from section 4.1 in the textbook.

Probability is a way of measuring the likelihood that some event of interest will occur. You see this everyday with things like weather forecasts (there is 10% chance of rain, etc). In this section, you will study some ideas for finding the more simple types of probabilities. Eventually, we will build up to using probability distributions in chapters 5 and 6.

Terminology

You should be comfortable with the following terminology for this chapter.

Probability Experiment (or just “experiment”) - For our purposes, an experiment is any activity where there is a measurable result.
Outcome - the result of a single trial (or run) of a probability experiment – also called “simple event”.
Sample Space - the set of ALL POSSIBLE OUTCOMES for a given experiment.
Event - a subset of the sample space (a collection of one of more outcomes).

Thinking about sample spaces

The sample space of any probability experiment shows all possibilities that could occur (all possible outcomes). For example, consider the following experiments and their sample spaces. Notice the set notation used. S = {}

Experiment Roll a single fair 6-sided die. Sample space S = {1,2,3,4,5,6}
Experiment Flip two fair coins. Sample space S = {HH, TT, HT, TH}

An event would be any selection of outcomes. For example, the event "a number larger than 3 is rolled" could be represented as E = {4, 5, 6}. Notice that this event has three outcomes.

Another event could be "both coins land on tails". This event could be represented as A = {TT}. Notice that this is a simple event, it is made up of only one outcome.

Some properties of events

When working with probability in chapter 4, it will be important to check if events satisfy certain properties. If they do, then our approach to the calculations may be different.

Independent vs. dependent events

If two events are independent, then the fact that one event occurs does not affect the probability that the other will occur. For example, the events "get an A in this course" and "you drive a red car" are independent (have no effect on each other).

As you will see later, many other less obvious events are also independent. Things like coin flips, die rolls, etc are always independent since the result of one flip or one roll does not affect what will happen the next time. Even if a coin lands on tails 100 times, the probability of tails on the 101st flip stays the same!

Mutually exclusive events

Mutually exclusive events cannot occur at the same time. For example, if you surveyed customers about their grocery shopping, the events "a customer spends more than $200 a week" and "a customer spends less than $100 a week" would be mutually exclusive since only one can be true for a given customer.

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