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Simple Probability Experiments

This lesson explains simple probability experiments, including trials, outcomes, and sample spaces, using coin tosses, dice rolls, and spinners as models, and shows how to apply the probability formula to real examples.

What is a simple probability experiment?

A simple probability experiment is any activity with a clear set of possible results, where you repeat the activity (a "trial") and record what happens. Flipping a coin, rolling a single die, spinning a spinner, or drawing one card from a deck are all simple probability experiments. Each one has a fixed, countable list of things that could happen, which makes them the easiest place to start when you are just understanding probability for the first time.

Before calculating anything, it helps to know three key words:

Trial: one single run of the experiment, such as one coin flip.
Outcome: one possible result of a trial, such as landing on heads.
Sample space: the list of every possible outcome, written as a set.

The probability formula

For a simple experiment where every outcome is equally likely, the probability of an event \( A \) is:

\( P(A) = \dfrac{n(A)}{n(S)} \)

Here \( n(A) \) is the number of outcomes that count as a "success" for event \( A \), and \( n(S) \) is the total number of outcomes in the sample space. Probability values always sit between \( 0 \) and \( 1 \), where \( 0 \) means impossible and \( 1 \) means certain.

Example 1: Tossing a coin

A fair coin has two equally likely outcomes: heads (H) and tails (T), so the sample space is \( S = \{H, T\} \) and \( n(S) = 2 \). If the event \( A \) is "landing on heads," then \( n(A) = 1 \), so:

\( P(H) = \dfrac{1}{2} \)

Sample space for one coin toss: two equally likely outcomes.

Example 2: Rolling one die

A standard die has six faces, so the sample space is \( S = \{1, 2, 3, 4, 5, 6\} \) and \( n(S) = 6 \). Suppose the event \( A \) is "rolling an even number." The favorable outcomes are \( \{2, 4, 6\} \), so \( n(A) = 3 \), giving:

\( P(A) = \dfrac{3}{6} = \dfrac{1}{2} \)

Sample space for one die roll; green squares are the even outcomes.

Example 3: Spinning a spinner

A spinner divided into 4 equal colored sections (red, blue, green, yellow) has \( n(S) = 4 \). The probability of landing on any one specific color is \( \dfrac{1}{4} \), and the probability of landing on either red or blue is \( \dfrac{2}{4} = \dfrac{1}{2} \), since two of the four sections satisfy that event.

Experimental versus theoretical probability

The formula above gives the theoretical probability, based only on counting outcomes. If you actually run the experiment, say tossing a coin 50 times and recording 27 heads, you get an experimental probability of \( \dfrac{27}{50} \). The more trials you run, the closer the experimental result usually gets to the theoretical value. This idea becomes important once you start predicting outcomes from repeated trials.

Putting it into words

Once you're comfortable finding sample spaces and applying the probability formula to a single coin, die, or spinner, the next step is combining those skills with real-world contexts, which is exactly what you'll practice in probability word problems.

Quick checklist for any simple probability experiment

List every possible outcome to build the sample space \( S \).
Count \( n(S) \), the total number of outcomes.
Identify which outcomes satisfy the event you care about, and count them as \( n(A) \).
Divide: \( P(A) = \dfrac{n(A)}{n(S)} \).

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