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9. Understanding Sampling
Sampling is a common application of the concepts of permutations and combinations. Learn how to calculate probabilities associated with sampling in this lesson, as well as the effect that replacement has on these probabilities.

Lesson Goal (00:15)
The goal of this lesson is to learn how to calculate probabilities associated with sampling.

Understanding the Problem (00:20)
Sampling is the process of selecting a permutation or combination of objects from a set. In our example, we have a company of employees spread across three offices. We want to understand the probability of selecting exactly one employee from each office if we select three employees at random.

Probability for a Combination (01:16)
Our aim is to find the probability for a combination, however we first find the probability for an individual permutation. We calculate the probability for one possible permutation where our selection of 3 people contains exactly one from each office. After we make each selection, we assume that the number of people from the relevant office and the number of people in the company as a whole both decrease by 1. This is known as sampling without replacement.
We find that the probability for each permutation is the same, so we find the probability for the combination by multiplying the probability for a permutation by the number of permutations in the combination of interest. This gives us the probability that a selection of 3 people will contain exactly one person from each office, if we sample without replacement.

Considering Replacement (03:49)
An alternative method of sampling is sampling with replacement. Here we assume that when a person is selected from a particular office, another person replaces them. As a result, the number of people available for subsequent picks from that office and from the company as a whole does not decrease.
When we calculate the probabilities of the same combination, but with replacement, we find the probability that our selection of 3 people will all come from different offices is lower than without replacement. This is because replacement increases the probability that second or subsequent picks will come from an office that has already been selected from previously. This demonstrates how probabilities can be altered depending on whether we sample with or without replacement.

Course Summary (05:52)
In this course, we covered the following:

Introduced fundamental probability principles

Calculated probabilities for one or more events

Learned how to use Bayes’ Theorem

Visualized probability problems

Calculated permutations and combinations
