Probability Without Replacement

Probability Without Replacement

When an item is not replaced, the total number of objects changes after each draw. That means the probability for the next step depends on what already happened.

1) List the stages

For each draw, write the probability of that draw happening. Use the updated total and updated number of favorable objects after every selection.

2) Multiply along the path

For a sequence of events, multiply the probabilities of each step. This gives the probability of the entire sequence.

3) Simplify carefully

Leave the answer as a fraction in simplest form unless another form is requested. Reduce common factors at the end.

4) Watch for order

If the exercise asks for a specific order, keep the order exactly as stated. If it asks for “at least” or “exactly,” count all valid cases and add their probabilities.

5) Check your result

A valid probability must be between 0 and 1. If your answer is greater than 1 or negative, something went wrong.

Quick example pattern

If you are selecting two items without replacement and want both to have a certain property, use:

• probability of the first draw having the property
• times probability of the second draw having the property after the first draw is removed

This method works for all without-replacement probability problems.