Wiki Page, Math 13 - Perez Sandbox : Module 11 - Another Look at Probability (1b of 2)

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2026-01-17T21:15:09-08:00

In summary, the probability that an event will occur is a number between (and including) 0 and 1. We write this idea in mathematical notation as .

EXAMPLE A closer look at how we calculate probabilities

You may have had some experience with probability using coins, cards, and dice.

What is the probability that when you flip a coin you get heads?

What is the probability that when you roll a die you get a 4?

We can answer these types of probability questions without collecting data. In situations where the outcomes are equally likely, we can use mathematics to calculate the probability instead of collecting data. For example, what is the probability of getting heads when you toss a fair coin? There are two equally likely outcomes: heads or tails. So P(heads) = 1/2. The probability of rolling a 4 when you toss a die, P(4), is 1/6 because there are 6 equally likely outcomes. These are examples of theoretical probability. No experiment or data collection is necessary.

But what if the coin is not a fair coin? If the tail side of the coin is lighter, then a coin toss is more likely to turn up as a tail. The P(tail) will be greater than 1/2 but we don't know the probability. Or what if we are tossing a bottle cap? Is the probability that it lands right-side-up 1/2? Who knows?

In these situations, we may not be able to determine a theoretical probability. Instead, we collect data and estimate the probability based on how often the outcome occurred in the data. This estimate is called an empirical probability.

We have already been using empirical probabilities. For example, we used a two-way table to estimate the risk of a fatality when not wearing a seatbelt. In that example we used data about car accidents to estimate a probability of a fatal outcome.

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