Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is [latex]\left\{1,2,3,4,5,6\right\}[/latex]. An event is any subset of a sample space.

The likelihood of an event is known as probability. The probability of an event [latex]p[/latex] is a number that always satisfies [latex]0\le p\le 1[/latex], where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like the table below.

Outcome Probability
Winning the raffle 1%
Losing the raffle 99%

The sum of the probabilities listed in a probability model must equal 1, or 100%.

How To: Given a probability event where each event is equally likely, construct a probability model.

  1. Identify every outcome.
  2. Determine the total number of possible outcomes.
  3. Compare each outcome to the total number of possible outcomes.

Example 1: Constructing a Probability Model

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

Solution

Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.

Assign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is [latex]\frac{1}{6}[/latex].

Outcome Roll of 1 Roll of 2 Roll of 3 Roll of 4 Roll of 5 Roll of 6
Probability [latex]\frac{1}{6}[/latex] [latex]\frac{1}{6}[/latex] [latex]\frac{1}{6}[/latex] [latex]\frac{1}{6}[/latex] [latex]\frac{1}{6}[/latex] [latex]\frac{1}{6}[/latex]

Q & A

Do probabilities always have to be expressed as fractions?

No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.

Try It 1

Construct a probability model for tossing a fair coin.

Solution

Computing Probabilities of Equally Likely Outcomes

Let [latex]S[/latex] be a sample space for an experiment. When investigating probability, an event is any subset of [latex]S[/latex]. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in [latex]S[/latex]. Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in [latex]S[/latex], so the probability of the event is [latex]\frac{4}{6}=\frac{2}{3}[/latex].

A General Note: Computing the Probability of an Event with Equally Likely Outcomes

The probability of an event [latex]E[/latex] in an experiment with sample space [latex]S[/latex] with equally likely outcomes is given by

[latex]P\left(E\right)=\frac{\text{number of elements in }E}{\text{number of elements in }S}=\frac{n\left(E\right)}{n\left(S\right)}[/latex]

[latex]E[/latex] is a subset of [latex]S[/latex], so it is always true that [latex]0\le P\left(E\right)\le 1[/latex].

Example 2: Computing the Probability of an Event with Equally Likely Outcomes

A number cube is rolled. Find the probability of rolling an odd number.

Solution

The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.

[latex]P\left(E\right)=\frac{3}{6}=\frac{1}{2}[/latex]

Try It 2

A number cube is rolled. Find the probability of rolling a number greater than 2.

Solution