Probability is the science of uncertainty.

• When we run an experiment, we are unsure of what the outcome will be.
• Because of this uncertainty, we say an experiment is a random process.

The probability of an event is the proportion of times it would occur if the experiment were run infinitely many times.

For a collection of equally likely events, this looks like: \[ \text{probability of event} = \frac{\text{number of ways event can occur}}{\text{number of possible outcomes}} \]

An event is some specified possible outcome (or collection of outcomes) we are interested in observing.

You want to roll a 6 on a six-sided die.

• There are six possible outcomes \(\{1,2,3,4,5,6\}\).
• We assume that each die face is equally likely to appear on a single roll of the die.
  • The die is fair.
• The probability of rolling a 6 is \[\frac{\text{number of ways to roll a 6}}{\text{number of possible rolls}} = \frac{1}{6}\]

We can extend this to a collection of events.

• The probability of rolling a 5 or a 6: \[\frac{\text{number of ways to roll a 5 or 6}}{\text{number of possible rolls}} = \frac{2}{6}\]

The collection of all possible outcomes is called a sample space, denoted \(S\).

• For the six-sided die, \(S=\{1,2,3,4,5,6\}\).

To simplify our writing, we use probability notation:

• Events are assigned capital letters.
• \(P(A)\) denotes the probability of event \(A\).
• Sometimes we will also shorten simple events to just a number.
  • For example, \(P(1)\) might represent “the probability of rolling a 1”.

We can estimate probabilities from a sample using a frequency distribution.

Example: Consider the frequency distribution.

If a student is selected at random, the probability of selecting a sophomore is \[\frac{10 \text{ sophomores }}{30 \text{ students}} \approx 0.3333\]

The probability of selecting a junior or a senior is \[\frac{3\text{ juniors }+5\text{ seniors }}{30\text{ students }} = \frac{8}{30} \approx 0.2667\]

Using probability notation, let \(A\) be the event we selected a junior and \(B\) be the event we selected a senior. Then \[P(A \text{ or } B) = \frac{8}{30} \approx 0.2667\]