Probability Notes

Published on: September 29, 2026

This Probability Calculator finds the chance of a single outcome and shows the working line by line. It divides the number of outcomes counting as a success by the total number of possible outcomes, assuming every outcome is equally likely. The answer always lands between zero and one, which is a quick way to catch a mistake in the counting. Following the same pattern every time makes probability easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two counts into the formula.
  4. Divide to get the probability.

Formula:

\(P = \frac{f}{n}\)
See Example 1 Hide Example 1

Problem

\(f = 2,\; n = 5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two counts into the formula.
  4. Divide to get the probability.

Step 1

Step 1 - Set up the problem.

In this problem: Out of \(5\) possible outcomes, \(2\) count as a success.

\(f = 2,\; n = 5\)

Step 2

Step 2 - Write the formula.

In this problem: Probability is the share of all the outcomes that count as a success, so the favourable count is divided by the total count.

\(P = \frac{f}{n}\)

Step 3

Step 3 - Substitute the two counts into the formula.

In this problem: Replacing \(f\) with \(2\) and \(n\) with \(5\) gives \(P = \frac{2}{5}\).

\(P = \frac{2}{5}\)

Step 4

Step 4 - Divide to get the probability.

In this problem: Dividing gives \(2 \div 5 = 0.4\).

\(P = \frac{2}{5} = 0.4\)

Final Answer

\(P = 0.4\)
See Example 2 Hide Example 2

Problem

\(f = 7,\; n = 9\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two counts into the formula.
  4. Divide to get the probability.

Step 1

Step 1 - Set up the problem.

In this problem: Out of \(9\) possible outcomes, \(7\) count as a success.

\(f = 7,\; n = 9\)

Step 2

Step 2 - Write the formula.

In this problem: Probability is the share of all the outcomes that count as a success, so the favourable count is divided by the total count.

\(P = \frac{f}{n}\)

Step 3

Step 3 - Substitute the two counts into the formula.

In this problem: Replacing \(f\) with \(7\) and \(n\) with \(9\) gives \(P = \frac{7}{9}\).

\(P = \frac{7}{9}\)

Step 4

Step 4 - Divide to get the probability.

In this problem: Dividing gives \(7 \div 9 \approx 0.78\).

\(P = \frac{7}{9} \approx 0.78\)

Final Answer

\(P \approx 0.78\)
See Example 3 Hide Example 3

Problem

\(f = 3,\; n = 8\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two counts into the formula.
  4. Divide to get the probability.

Step 1

Step 1 - Set up the problem.

In this problem: Out of \(8\) possible outcomes, \(3\) count as a success.

\(f = 3,\; n = 8\)

Step 2

Step 2 - Write the formula.

In this problem: Probability is the share of all the outcomes that count as a success, so the favourable count is divided by the total count.

\(P = \frac{f}{n}\)

Step 3

Step 3 - Substitute the two counts into the formula.

In this problem: Replacing \(f\) with \(3\) and \(n\) with \(8\) gives \(P = \frac{3}{8}\).

\(P = \frac{3}{8}\)

Step 4

Step 4 - Divide to get the probability.

In this problem: Dividing gives \(3 \div 8 \approx 0.38\).

\(P = \frac{3}{8} \approx 0.38\)

Final Answer

\(P \approx 0.38\)
See Example 4 Hide Example 4

Problem

\(f = 5,\; n = 12\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two counts into the formula.
  4. Divide to get the probability.

Step 1

Step 1 - Set up the problem.

In this problem: Out of \(12\) possible outcomes, \(5\) count as a success.

\(f = 5,\; n = 12\)

Step 2

Step 2 - Write the formula.

In this problem: Probability is the share of all the outcomes that count as a success, so the favourable count is divided by the total count.

\(P = \frac{f}{n}\)

Step 3

Step 3 - Substitute the two counts into the formula.

In this problem: Replacing \(f\) with \(5\) and \(n\) with \(12\) gives \(P = \frac{5}{12}\).

\(P = \frac{5}{12}\)

Step 4

Step 4 - Divide to get the probability.

In this problem: Dividing gives \(5 \div 12 \approx 0.42\).

\(P = \frac{5}{12} \approx 0.42\)

Final Answer

\(P \approx 0.42\)