Replacement

Definition

Replacement is the practice of returning a selected item back into the pool of possible selections before making another choice. This affects the probability calculations as each selection is independent and does not alter the total number of items available for future selections.

5 Must Know Facts For Your Next Test

1. With replacement, each selection is an independent event because the composition of the sample space remains unchanged.
2. The probability of selecting any item remains constant throughout multiple trials when using replacement.
3. In scenarios without replacement, each selection would alter the sample space and affect subsequent probabilities.
4. Replacement ensures that events are not mutually exclusive since each item can be chosen more than once over multiple trials.
5. Calculations involving replacement often use binomial or multinomial distributions to determine probabilities.

Review Questions

• How does the concept of replacement affect the independence of events in a probability experiment?
• What changes in probability calculations occur when selecting items with replacement versus without replacement?
• Give an example where using replacement would be necessary to maintain equal probabilities across trials.

Related terms

Independent Events: Two or more events are independent if the occurrence or non-occurrence of one does not affect the occurrence of others.

Mutually Exclusive Events: Events are mutually exclusive if they cannot happen at the same time; i.e., if one event happens, it precludes others from occurring.

Binomial Distribution: $$\text{A probability distribution that summarizes the likelihood that a value will take one of two independent states under a given number of parameters.}$$