Dependent or Independent?

Before applying any multiplication rule to a probability problem, decide whether the events involved are independent or dependent. This lesson practices reading a problem's wording for replacement and shared-pool language, justifying that decision, and then applying the matching multiplication rule.

What you will learn

• Decide whether two events in a story are independent or dependent
• Justify the decision from the problem's wording
• Apply the correct multiplication rule once independence is decided

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Reading the setup for replacement language

Some problems state the replacement rule directly. Phrases like "with replacement" or "the item is put back before the next draw" describe independent events, since the pool returns to its original state before the next draw happens. Phrases like "without replacement," "the item is not returned," or "from what remains" describe dependent events, since the pool has permanently changed.

Spotting this language first, before doing any arithmetic, prevents applying the wrong multiplication rule. When such a phrase appears explicitly, trust it over any assumption about how a typical experiment usually works.

Reading the setup when replacement isn't stated outright

Many problems never use the word "replacement" at all, so the decision has to come from the structure of the setup. Two separate, non-interacting sources — a different bag for each event, two different classrooms, two different days, or two different devices — are independent, because there is no shared pool for one outcome to change.

By contrast, any situation where several items are taken from one shared, fixed group, and none of them return to that group, is dependent, even if the word "probability" or "replacement" never appears. Selecting several officers one at a time from the same club, or dealing several cards from the same deck, are both dependent because each selection permanently shrinks the same pool the next selection draws from.

Applying the right rule once independence is decided

Once a problem is identified as independent, every stage's probability stays exactly as it was originally, so the final answer is simply the product of the unchanged probabilities. Once a problem is identified as dependent, the count and category totals must be updated after every stage before computing the next fraction, and only then should the updated stage probabilities be multiplied together.

Writing out the justification in a sentence — "these are independent because..." or "these are dependent because..." — before doing the arithmetic catches mistakes early and makes the final multiplication straightforward to check.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Assuming an event is dependent just because two draws happen from bags with the same kind of items, without checking whether the pools are actually shared.
• Missing the phrase "without replacement" or "not returned" and treating a dependent setup as independent.
• Updating the counts for a second stage even when the two events come from entirely separate pools that were never connected.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Look for replacement language: "with replacement" or separate, non-interacting setups signal independence, while "without replacement" or a shared shrinking pool signals dependence.
• Independent events keep the same probability at every stage; dependent events require updating counts after each stage.
• Justify the independence decision explicitly before choosing which multiplication rule to apply.