lesson
Count equally likely outcomes carefully
1 · Learn
A probability lies between 0 and 1. For equally likely outcomes, probability is favourable outcomes divided by all outcomes. A fair six-sided die has six equally likely faces in the model; an unfair device may not.
Relative frequency from experiments estimates probability but can vary, especially with few trials. A short run of one outcome does not guarantee the next result will compensate.
2 · Worked example
For a fair die, rolling a number greater than 4 has probability 2/6 = 1/3 because 5 and 6 are the favourable faces.
3 · Your turn
For a fair six-sided die, find the probability of an even number. Does rolling three odd numbers guarantee the next is even?
Check your answer
3/6 = 1/2. No; the next independent roll is not forced to balance earlier results.
4 · Apply your learning
Record a safe simulated experiment and compare relative frequency with the theoretical model, noting random variation.
Adult guidance and safety
Require a reasoned method and an independent check. Use accurate number lines, geometric drawings and data displays where the task needs them. These text models require varied practice and diagnostic assessment before curriculum approval.
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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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