lesson
Multiply along branches and add disjoint routes
1 · Learn
A probability tree records conditional stages. Multiply probabilities along one complete route. Add probabilities for different mutually exclusive routes leading to the event of interest.
Without replacement, the second-stage denominator and sometimes numerator change. With replacement, the original probabilities return. Label every branch and check branches from each node sum to one.
A theoretical model predicts long-run behaviour, not an exact short-run count. Empirical proportions tend to stabilise with larger unbiased samples, but random variation remains.
2 · Worked example
A bag has 3 red and 2 blue. Without replacement, P(RR)=3/5×2/4=3/10. P(one of each)=3/5×2/4 + 2/5×3/4=3/5.
3 · Your turn
A bag has 4 red and 2 blue. Two are drawn without replacement. Find P(two red).
Check your answer
4/6×3/5=2/5.
4 · Apply your learning
Draw both without-replacement and with-replacement trees for the same bag. Compare the second-stage branches and predictions.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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