Independent or Dependent Events

1) Decide whether the events affect each other

Two events are independent if the outcome of one does not change the probability of the other. They are dependent if the first event changes what is possible, or changes the probability, for the second event.

2) Read the situation carefully

Ask yourself:

• Is anything being removed or not replaced?
• Does the second event happen after the first in a way that changes the sample space?
• Are the events from separate actions where one does not influence the other?

If the answer involves a changing total, changing counts, or no replacement, the events are usually dependent.

3) Use the appropriate probability rule

• For independent events: multiply the probabilities of the events.
• For dependent events: multiply the first probability by the updated probability for the second event.

Write each probability as a fraction, decimal, or percent as required, then simplify the final result.

4) Check your answer

Your final probability should be between 0 and 1. If the events are dependent, make sure the second probability matches the new situation after the first event. If the events are independent, the probability of one event should not have changed the other.

5) Common mistake to avoid

Do not reuse the first probability for the second event when the situation changes. Always update the numbers before multiplying.