Hard

Applications to Probability and Statistics

Simple Explanation

Permutations and combinations are essential tools for computing probabilities: the probability of a specific outcome is often (favorable outcomes) / (total outcomes), where both counts are found using permutation or combination formulas — consistently, using the SAME method for both.

Why Do We Need It?

This connects the abstract counting techniques of this chapter to genuinely practical probability and statistics calculations used across science, engineering, and everyday decision-making.

Formula

Combinations of r Objects from n

C(n,r) = n! / [r!(n−r)!]

The number of ways to choose r objects from a total of n distinct objects, where the ORDER of selection does not matter.

n: — the total number of distinct objects available
r: — the number of objects being chosen, r ≤ n

When to use it: Whenever a group or subset is being chosen and the order within the group does not matter (like choosing a committee).

Worked Example

Use combinations to find a probability

A jar has 5 red and 3 blue marbles (8 total). If 2 marbles are drawn at random without replacement, find the probability both are red.

Why Does This Work?

Since the marbles are drawn without regard to order, EVERY count involved (both the total ways to draw 2 marbles, and the favorable ways to draw 2 red ones) must consistently use combinations — mixing permutations and combinations between the numerator and denominator would give a wrong probability.

Real-Life Example

Quality control sampling in manufacturing

A quality inspector randomly selects a sample of items from a batch and wants the probability that the sample contains a certain number of defective items.

This is exactly the same type of calculation — combinations give both the total number of possible samples and the number of "favorable" samples, from which the probability is computed.

Practice

A jar has 4 red and 6 blue marbles (10 total). Find the number of ways to choose 2 blue marbles (order does not matter).

Hard

Common mistake

Mixing permutations and combinations inconsistently between the numerator and denominator of a probability calculation — both must use the SAME counting method throughout.

Quick Review

Probability = (favorable outcomes) / (total outcomes), using permutation or combination counts.
Use the SAME counting method (usually combinations, when order does not matter) throughout one calculation.
This connects the chapter's counting techniques directly to probability and statistics.