In probability, a selection is the action of choosing one or more items from a set or group. We can perform selections either with replacement or without replacement.

In selection with replacement, each chosen item is returned to the original group before the next choice, keeping the probabilities constant across selections.

In selection without replacement, the chosen items are removed from the group, causing probabilities to change after each pick because the number of available items decreases.

This difference significantly impacts how probabilities are calculated, especially in problems involving multiple selections.

Example

Julia selects 3 marbles from a bag containing 3 blue marbles and 6 yellow marbles. Find the probability that Julia selects 3 blue marbles if

(a) the marbles are replaced after each selection,

(b) the marbles are not replaced.

If the marbles are replaced, the probability of selecting a blue marble is always 3+63​=31​. So the probability of selecting 3 blue marbles is (31​)3=271​.

If the marbles are not replaced, then the probability of selecting a blue marble is

• 93​=31​ the first time,

  
  

• 82​=41​ the second time, and

  
  

• 71​ the third time.

Multiplying all of these gives the probability 841​.