Sampling with replacement is used to find **probability with replacement**. In other words, you want to find the probability of some event where there’s a number of balls, cards or other objects, and you replace the item each time you choose one.

**Example**: Let’s say you had a population of 7 people, and you wanted to sample 2. Their names are:

- John
- Jack
- Qiu
- Tina
- Hatty
- Jacques
- Des

You could put their names in a hat. If you **sample with replacement**, you would choose one person’s name, put that person’s name back in the hat, and then choose another name. The possibilities for your two-name sample are:John, John

- John, Jack
- John, Qui
- Jack, Qui
- Jack Tina
- …and so on.

When you sample with replacement, your two items are independent. In other words, one does not affect the outcome of the other. You have a 1 out of 7 (1/7) chance of choosing the first name and a 1/7 chance of choosing the second name.

- P(John, John) = (1/7) * (1/7) = .02.
- P(John, Jack) = (1/7) * (1/7) = .02.
- P(John, Qui) = (1/7) * (1/7) = .02.
- P(Jack, Qui) = (1/7) * (1/7) = .02.
- P(Jack Tina) = (1/7) * (1/7) = .02.

Note that P(John, John) just means “the probability of choosing John’s name, and then John’s name again.” You can figure out these probabilities using the multiplication rule.