Math Core

A weather app, a board game designer and a sports announcer all use the same tool: a probability model. A model lists everything that can happen and gives each outcome a probability, so you can answer any question about the situation in one place.

What is a probability model?

Definition

Probability model

A probability model is a sample space together with a probability for each outcome. In every probability model:

each probability is between $0$ and $1$ , and
the probabilities of all the outcomes add up to exactly $1$ .

A model is often shown as a table. Here is the model for one roll of a fair number cube.

Outcome	$1$	$2$	$3$	$4$	$5$	$6$
Probability	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$

The six probabilities add to $\dfrac{6}{6} = 1$ , as they must. To find the probability of an event, add the probabilities of its outcomes. For example, $P(1 \text{ or } 2) = \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{1}{3}$ .

Uniform models

Uniform and non-uniform models

In a uniform model, every outcome has the same probability. If there are $n$ outcomes, each has probability $\dfrac{1}{n}$ .

In a non-uniform model, some outcomes are more likely than others.

Worked example: A uniform model

A teacher writes the names of all $25$ students in her class on slips of paper and draws one at random. There are $13$ girls and $12$ boys. Describe the model, then find the probability that a girl's name is drawn.

Each student is one outcome, and every slip is equally likely, so the model is uniform: each student has probability $\dfrac{1}{25}$ .

The event "a girl is drawn" contains $13$ outcomes, so

P(\text{girl}) = 13 \times \frac{1}{25} = \frac{13}{25}.

Non-uniform models

Not every situation is fair. A spinner with sections of different sizes gives a non-uniform model.

Worked example: A spinner with unequal sections

A spinner is half red, one quarter blue and one quarter green. Write a probability model and find $P(\text{not red})$ .

The size of each section is its probability.

Outcome	Red	Blue	Green
Probability	$\dfrac{1}{2}$	$\dfrac{1}{4}$	$\dfrac{1}{4}$

Check: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{4} = 1$ . Then $P(\text{not red}) = \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2}$ .

Common mistake

Three outcomes do not mean each has probability $\dfrac{1}{3}$ . The spinner above has three colors, but red is twice as likely as blue. Use a uniform model only when you have a good reason to think the outcomes are equally likely.

Finding a missing probability

Because the probabilities must add to $1$ , you can find one that is missing.

Worked example: The missing piece

A bag of mixed nuts is used for a game. The table shows the probability of pulling out each kind.

Nut	Peanut	Almond	Cashew	Walnut
Probability	$0.45$	$0.25$	?	$0.1$

Find $P(\text{cashew})$ .

The known probabilities add to $0.45 + 0.25 + 0.1 = 0.8$ . The missing one is $1 - 0.8 = 0.2$ .

Building a model from data

Sometimes you can't count your way to the probabilities. A paper cup can land on its side, right side up or upside down, and there's no reason to think these are equally likely. Instead, you collect data and use the experimental probabilities as the model.

Worked example: Tossing a paper cup

Kai tosses a paper cup $100$ times.

Landing	On its side	Right side up	Upside down
Times	$68$	$22$	$10$

Build a probability model, then predict how many times the cup lands on its side in $250$ tosses.

Divide each count by $100$ :

Landing	On its side	Right side up	Upside down
Probability	$0.68$	$0.22$	$0.10$

The model is non-uniform, and the probabilities add to $1$ . The prediction is $0.68 \times 250 = 170$ times on its side.

Tip

When you build a model from data, divide by the total number of trials. Your probabilities will then add to $1$ automatically, which is a quick way to check your arithmetic.

Practice

Practice 1

A spinner lands on A, B, C or D. The probabilities are $P(\text{A}) = 0.2$ , $P(\text{B}) = 0.35$ and $P(\text{D}) = 0.15$ . What is $P(\text{C})$ ?