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All courses Math Foundations · MF.DP.3 57 of 60
Compute and compare theoretical and experimental probability in simple settings.

Find theoretical probability

Problem
A fair number cube has faces \(1~\text{through}~6\). Calculate the theoretical probability of rolling a number greater than \(4\) as a fraction.
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Hint

List all 6 possible rolls on the number cube, then identify which rolls are greater than 4.

Theoretical probability = favorable outcomes ÷ total equally likely outcomes. For a fair number cube, the denominator comes from all possible rolls.

Solution walkthrough

01

Use theoretical probability

\[P(\text{event})=(\text{favorable}~\text{outcomes})/(\text{equally}~\text{likely}~\text{outcomes})\]

A fair number cube makes each of its six faces equally likely.

02

List the sample space

\[{1,2,3,4,5,6}:~6~\text{outcomes}\]

The denominator is 6 because the cube has faces 1 through 6.

03

Count favorable outcomes

\[\text{numbers}~\text{greater}~\text{than}~4:~{5,6}:~2~\text{outcomes}\]

Only 5 and 6 satisfy the condition greater than 4.

04

Form and simplify the fraction

\[P=2/6=1/3\]

Divide numerator and denominator by 2 to write the probability in simplest form.

05

Report the probability

\[1/3\]

The theoretical probability of rolling a number greater than 4 is one third.

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Another way

  1. Two of the six faces work, so one out of every three equally sized pairs of faces is favorable.

!

Common mistake

Do not count 4 as favorable; greater than 4 means strictly above 4.

Solution walkthrough video