Math Core

There are two ways to find a probability. You can think it through by counting outcomes, or you can try it many times and see what actually happens. Knowing both, and how they are connected, lets you make predictions and test whether a game is fair.

Two kinds of probability

In the last lesson you found probabilities by counting equally likely outcomes. That kind of probability is called theoretical. Now you'll find probabilities from data.

Definition

Experimental probability

The theoretical probability of an event is what you expect from counting:

P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}.

The experimental probability of an event is what actually happened in trials:

P(\text{event}) \approx \frac{\text{number of times the event happened}}{\text{total number of trials}}.

Each repeat of the experiment (one flip, one roll, one spin) is a trial.

Experimental probability is also called relative frequency: the frequency of the event relative to the number of trials.

Worked example: Comparing the two

Jada rolls a number cube $50$ times. She rolls a $3$ twelve times. Find the experimental and theoretical probabilities of rolling a $3$ .

Experimental: $\dfrac{12}{50} = \dfrac{6}{25} = 0.24$ .
Theoretical: one of the $6$ faces is a $3$ , so $\dfrac{1}{6} \approx 0.17$ .

The two numbers are close but not equal. That is normal. Chance experiments rarely match the theory exactly.

More trials, better estimates

Here is what happened when a fair coin was flipped a growing number of times.

Flips	Heads	Experimental $P(\text{heads})$
$10$	$7$	$0.7$
$100$	$46$	$0.46$
$1{,}000$	$508$	$0.508$
$10{,}000$	$4{,}981$	$0.4981$

With only $10$ flips, $0.7$ is far from the theoretical $0.5$ . As the number of trials grows, the experimental probability settles closer and closer to the theoretical probability.

The link between them

When you repeat an experiment many times, the experimental probability tends to get close to the theoretical probability. A few trials can be far off; many trials usually aren't.

This also works in reverse. If you run a lot of trials and the results are still far from what you expected, the theory might be wrong. Maybe the coin is bent, the number cube is weighted, or the spinner sticks.

Making predictions

A probability tells you what fraction of the time an event should happen. To predict how many times, multiply.

\text{expected number of times} = P(\text{event}) \times \text{number of trials}

Worked example: Predicting with theoretical probability

A spinner has $4$ equal sections: red, blue, green and yellow. If you spin it $80$ times, about how many times should it land on blue?

$P(\text{blue}) = \dfrac{1}{4}$ , so the prediction is $\dfrac{1}{4} \times 80 = 20$ times.

You probably won't get exactly $20$ , but you should get a number near $20$ .

Worked example: Predicting with experimental probability

Some events can't be found by counting. A factory tests $150$ phone chargers and finds that $6$ are faulty. About how many faulty chargers would you expect in a shipment of $2{,}000$ ?

The experimental probability of a faulty charger is $\dfrac{6}{150} = \dfrac{1}{25} = 0.04$ .

The prediction is $0.04 \times 2{,}000 = 80$ faulty chargers.

Common mistake

A prediction is an estimate, not a promise. If you flip a fair coin $10$ times and get $7$ heads, the coin is not "due" for tails on the next flip. Each flip still has a $\dfrac{1}{2}$ chance of heads. The experimental probability evens out over many more flips, not by making up for the past.

Tip

Before you multiply, ask whether your prediction makes sense. The expected number can never be more than the number of trials, and it should be about half the trials when the probability is about $\dfrac{1}{2}$ .

Practice

Practice 1

Luis made $18$ of his $25$ free throws this season. What is the experimental probability that he makes a free throw? Give your answer as a decimal.