lesson
The second probability can change
1 · Learn
When an item is drawn without replacement, both the contents and total of the collection change. A probability tree must use the correct conditional probabilities on each second branch.
Multiply along a branch for a joint outcome, then add mutually exclusive routes for the required event. Do not use the original denominator on every draw.
2 · Worked example
From 3 red and 2 blue counters, two reds without replacement has probability (3/5)(2/4) = 3/10.
3 · Your turn
For that same bag, find the probability of two blue counters without replacement.
Check your answer
(2/5)(1/4) = 1/10.
4 · Apply your learning
Complete all four colour branches and verify their probabilities total one.
Adult guidance and safety
Require reasoning and a check. Provide accurate graphs, construction instruments and varied practice. Annual KS3 allocation is proposed; do not treat these short models as a complete assessed programme.
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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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