Construct Probability Models

Learning Outcomes

  • Construct a probability model that assigns the probability of each outcome in a sample space.
  • Compute the probability of an event with equally likely outcomes.

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.

tip for success

Be sure to try the examples and practice problems in this section on paper to build up your intuition for the next section’s material.

Example: Constructing a Probability Model

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

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

Construct a probability model for tossing a fair coin.

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)=\dfrac{\text{number of elements in }E}{\text{number of elements in }S}=\dfrac{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: Computing the Probability of an Event with Equally Likely Outcomes

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

Try It

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