Independent and Dependent Events

What you will learn

• Recognise independent events.
• Recognise dependent events.
• Use multiplication for independent events.
• Update counts for dependent events.
• Explain replacement and no replacement.
• Use tree-branch wording to justify combined probabilities.
• Combine route probabilities correctly when more than one order gives the required outcome.
• Use complements or route pairs when probability wording says at least one, same colour or exactly one.

Key facts before you start

Independent and dependent events infographic

Why this matters

Choosing independent or dependent affects every probability calculation after the first event. It is the difference between keeping the same fraction and updating the total.

Prior knowledge

You should already be comfortable with:

• Probability trees.
• Fractions.
• Multiplying probabilities.

Clear explanation

Main idea

Events are independent if one happening does not change the probability of the other. Tossing a coin twice is independent because the second toss is still heads or tails with probability 1/2.

Method

Events are dependent if the first event changes the probability of the second. Taking counters without replacement is dependent because the bag has fewer counters after the first pick.

Independent does not mean equally likely. It only means the probability for the next event has not been changed by what happened before.

For combined events, multiply along the route after deciding whether the second probability stays fixed or changes. If several routes work, multiply each route first, then add the route probabilities.

Words such as exactly one, either order or same colour often mean there is more than one successful route. Write each route separately so you do not add raw branch fractions or miss an order.

For at least one wording, the complement is often shorter: calculate the probability of no successes, then subtract from 1.

Answer tip

Replacement usually makes picks independent; no replacement usually makes them dependent. In a two-stage question, write the first probability, then ask: has the total or the number of successful items changed?

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For independent and dependent events, marks usually come from identifying whether probabilities stay fixed or change, showing route multiplication, updating totals for no-replacement contexts and using complements or route addition where the wording requires it.

Common mistakes

• Treating without replacement as independent: after an item is removed, the second denominator is usually smaller.
• Changing both counts when only one should change: if the first counter is red, the number of blue counters may stay the same but the total changes.
• Forgetting what AND means: for one route with two events, multiply the probabilities along that route.
• Adding separate routes too soon: multiply each successful route first, then add routes only when there is more than one way to succeed.

Extension challenge

Explain the difference between rolling a fair dice twice and taking two counters from a bag without replacement. Give one probability calculation for each situation.

Exam-board guidance

Independent and Dependent Events appears within the shared GCSE Maths probability content used by the supported routes. Exact wording, tiering and calculator expectations can vary, but the core skill is the same: decide whether the first event changes the second probability, show the multiplication and check the context.

Next lesson

Next, continue with Conditional Probability.