Understanding the concepts of “with replacement” and “without replacement” is crucial for solving many probability problems. Here’s a general overview of each:
With Replacement
When we sample with replacement, after selecting an element from the sample space, it is returned to the space before the next selection. This means that the same element can be chosen more than once, and the probability of each selection remains constant throughout the process.
Example: Imagine you have a jar with 3 red balls and 2 blue balls. If you pick a ball, note its color, and then return it to the jar before picking again, you are sampling with replacement. The probabilities remain:
- Probability of picking a red ball: 3/5
- Probability of picking a blue ball: 2/5
Since the ball is replaced each time, these probabilities do not change.
Without Replacement
When we sample without replacement, after selecting an element from the sample space, it is not returned before the next selection. This means that the same element cannot be chosen more than once, and the probabilities change with each selection because the sample space is reduced.
Example: Using the same jar with 3 red balls and 2 blue balls, if you pick a ball, note its color, and do not return it to the jar, you are sampling without replacement. The probabilities change after each pick:
First Pick:
- Probability of picking a blue ball: 2/5
- Probability of picking a red ball: 3/5
Second Pick (assuming the first ball was blue):
- Now, there are 4 balls left (3 red, 1 blue).
- Probability of picking a red ball: 3/4
- Probability of picking the remaining blue ball: 1/4
Second Pick (assuming the first ball was red):
- Now, there are 4 balls left (2 red, 2 blue).
- Probability of picking a red ball: 2/4=1/2
- Probability of picking a blue ball: 2/4=1/2
Without replacement, the probabilities adjust because the number of total balls in the jar decreases with each pick.
