Combined Probability
This topic covers 5 learning steps, guiding your child from the basics through to confident problem-solving. Each step includes a worked example and adaptive practice questions.
What Your Child Will Learn
- Two events (independent)
P(A and B) = P(A) × P(B) - Tree diagrams
Draw and read tree diagrams - With or without replacement
Understand dependent events - Expected frequency
Calculate expected outcomes - Challenge problems
Multi-event probability problems
Common Mistakes
- Multiplying probabilities when events are NOT independent (e.g. picking two sweets from a bag without replacement and using the same denominator both times)
Without replacement, the second probability changes because the total has reduced. If you have 3 red and 7 blue sweets (10 total), P(1st red) = 3/10, but P(2nd red) = 2/9, not 2/10, because one sweet has already been removed. - Adding probabilities for "and" events instead of multiplying (e.g. P(heads AND 6) = 1/2 + 1/6 instead of 1/2 × 1/6)
For "AND" (both events happening), MULTIPLY the probabilities. For "OR" (either event happening), ADD them. P(heads AND rolling 6) = 1/2 × 1/6 = 1/12.
Tips for Parents
- Draw tree diagrams together: list each event as a set of branches, write the probability on each branch, then multiply along branches for combined probabilities.
- Use a coin and a die: "What is the probability of getting heads AND a 6?" Draw the tree, multiply 1/2 × 1/6 = 1/12. Then try it 60 times and see how close you get to 5.
- Practise the "with and without replacement" difference using a bag of coloured counters. Remove one, ask how the probabilities change, then compare to putting it back.
- Teach the AND/OR rule: AND means multiply (both must happen, so it gets less likely), OR means add (either can happen, so it gets more likely).
Key Words
- Independent events — Events where one does not affect the other — flipping a coin does not affect rolling a die.
- Dependent events — Events where one affects the other — taking a sweet from a bag changes the probabilities for the next pick.
- Tree diagram — A branching diagram showing all possible outcomes of two or more events, with probabilities on each branch.
- With replacement — Putting an item back before choosing again — the probabilities stay the same each time.
- Without replacement — Not putting an item back — the total changes, so the probabilities change for the next pick.
- Expected frequency — The number of times you expect something to happen — probability × number of trials.
Where This Fits
Before this topic: Children should understand basic probability as a fraction, be able to list outcomes, and know the probability scale from 0 to 1.
After this topic: Combined probability leads to Venn diagrams, conditional probability, the addition and multiplication rules, and probability distributions at GCSE and A-level.
How MathCraft Teaches This
In MathCraft, Combined Probability is taught through the Money, Data & Measure adventure track. Your child follows guided lessons with friendly characters, works through examples step by step, then practises with questions that adapt to their level.
The adaptive engine tracks mastery across all 5 steps, revisiting concepts your child finds tricky and advancing when they're ready. Parents can see detailed progress in the Parent Dashboard.
Practise Combined Probability with MathCraft
Step-by-step lessons, worked examples, and adaptive practice — all wrapped in an adventure game your child will love.
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