Math Core

Will it rain tomorrow? Will your free throw go in? Probability is a number that measures how likely something is to happen. In this lesson you'll find probabilities by counting outcomes, compare them with what actually happens in an experiment, and use equations to answer "how many?" questions.

Outcomes and events

When you roll a number cube, the possible results are $1, 2, 3, 4, 5$ and $6$ . Each result is an outcome, and the list of all of them is the sample space. An event is any group of outcomes you care about, such as "rolling an even number," which is the outcomes $2$ , $4$ and $6$ .

Definition

Probability of an event

If all outcomes are equally likely, the probability of an event $A$ is

P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes}}.

The outcomes in $A$ are often called the favorable outcomes.

Worked example: Rolling a number cube

You roll a fair number cube. What is the probability of rolling a number greater than $4$ ?

The sample space is $1, 2, 3, 4, 5, 6$ : $6$ outcomes. The numbers greater than $4$ are $5$ and $6$ : $2$ outcomes.

P(\text{greater than } 4) = \frac{2}{6} = \frac{1}{3}.

The probability scale

A probability is always between $0$ and $1$ , and you can write it as a fraction, a decimal or a percent.

$P = 0$ : the event is impossible (rolling a $7$ on a number cube).
$P = \dfrac{1}{2}$ : the event is equally likely to happen or not (a coin landing heads).
$P = 1$ : the event is certain (rolling a number less than $7$ ).

Impossible at 0, certain at 1

The closer a probability is to $1$ , the more likely the event.

The complement

Every outcome either is in event $A$ or isn't. The event "not $A$ " is called the complement of $A$ . Since the two together cover every outcome, their probabilities add to $1$ .

Complement rule

P(\text{not } A) = 1 - P(A)

Worked example: Marbles in a bag

A bag holds $5$ red, $3$ blue and $2$ green marbles. You pick one without looking.

What is $P(\text{blue})$ ?
What is $P(\text{not blue})$ ?

Solutions. There are $5 + 3 + 2 = 10$ marbles.

$P(\text{blue}) = \dfrac{3}{10}$ .
$P(\text{not blue}) = 1 - \dfrac{3}{10} = \dfrac{7}{10}$ . You can check by counting: the $5$ red and $2$ green marbles make $7$ of the $10$ .

Experimental probability

So far you've found theoretical probability, which comes from counting equally likely outcomes. Experimental probability comes from actually doing the experiment:

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

The two often don't match exactly, but they usually get closer as the number of trials grows. Either kind can be used to make a prediction: multiply the probability by the number of trials.

Worked example: Spinning a spinner

A spinner has $4$ equal sections: red, blue, green and yellow. Lena spins it $50$ times and it lands on blue $18$ times.

What is the experimental probability of blue? What is the theoretical probability?
Using the theoretical probability, how many times would you expect blue in $200$ spins?

Solutions.

Experimental: $\dfrac{18}{50} = \dfrac{9}{25} = 0.36$ . Theoretical: $1$ blue section out of $4$ , so $\dfrac{1}{4} = 0.25$ .
$\dfrac{1}{4} \times 200 = 50$ times.

Getting $18$ blues instead of about $12.5$ in $50$ spins can happen just by chance. Results from a small number of trials vary a lot.

Common mistake

A probability is a long-run expectation, not a promise. A probability of $\dfrac{1}{4}$ does not mean that exactly one of every four spins lands on blue. It means that over many spins, about one-fourth of them will.

Using an equation

When the number of outcomes is unknown, write an equation.

Worked example: How many marbles?

A bag has $4$ red marbles and some blue marbles. The probability of picking red is $\dfrac{1}{3}$ . How many blue marbles are in the bag?

Let $t$ be the total number of marbles. Then $\dfrac{4}{t} = \dfrac{1}{3}$ . Cross-multiply: $t = 12$ . There are $12$ marbles in all, so $12 - 4 = 8$ are blue.

Check: $\dfrac{4}{12} = \dfrac{1}{3}$ . ✓

Tip

After finding a probability, ask if it makes sense. If more than half the marbles are red, $P(\text{red})$ should be greater than $\dfrac{1}{2}$ .