• Calculate the probability of an event in a chance experiment.
• Recognize the differences between theoretical and empirical probability.

What is a Probability?

The probability of an event is a number between [latex]0[/latex] and [latex]1[/latex]. What does this number tell us about the likelihood of an event occurring?

What is the probability that when you flip a coin you get heads? There are two equally likely outcomes: Heads or tails. So, [latex]P(\text{heads})=\frac{1}{2}[/latex].

This is the theoretical probability of getting heads when you toss a coin. We determine the number of ways an event can occur and divide by the total number of possible outcomes. No experiments or data collection is necessary.

What is the probability that a community college student is a first-generation student? Like tossing a coin, we also have two outcomes: First-generation student and not first-generation student.

But, is [latex]P(\text{first-generation student}) = \frac{1}{2}[/latex]? To estimate this probability, we have to collect data and cannot conclude that the percentage is [latex]50\%.[/latex] This is an example of empirical probability. Empirical probability of an event is an data-driven estimate of the likelihood of the event happening.

What is the Difference between Theoretical and Empirical Probability?

The two ways of determining probabilities are empirical and theoretical.

• The empirical probability is a probability gained from performing an experiment. The probability of an event is approximated by the relative frequency of the event. The empirical probability, calculated from a chance experiment, will be closer to the true probability the more times we repeat the chance experiment. Therefore, we can make probability statements only about chance (random) events.
• Theoretical probability comes into play when there is no experiment to perform. We assume that the outcomes of an event are all equally likely to occur. The theoretical probability is a long-run probability.

A Chance Event

When we say that an event is random or due to chance, we mean that the event is unpredictable in the short run but has a regular and predictable behavior in the long run.

A single flip of a coin has an uncertain outcome. We do not know if we will get heads or tails. If we flip the coin [latex]10[/latex] times, we are not guaranteed to get [latex]5[/latex] heads and [latex]5[/latex] tails. So, what exactly does it mean when we say [latex]P(\text{heads}) = 0.5[/latex]? The empirical probability (probability through an experiment) will approach the theoretical probability of [latex]0.5[/latex] after a large number of repetitions of coin flips. In some situations, such as flipping an unfair coin, we cannot calculate the theoretical probability. In these cases, we have to depend on data.

There is less variability in a large number of repetitions. This means that in the long run, we will see a pattern, so we are more confident about estimating the probability of an event using empirical probability with a large number of repetitions.

So, we cannot predict whether an individual toss will be heads, but in the long run, the outcomes have a predictable pattern. The relative frequency of heads is very close to [latex]0.5[/latex] for a fair coin.

The main point is that the probability of an event [latex]A[/latex] is the relative frequency with which that event occurs in a long series of repetitions.