Rukasha Maths

GCSE maths Beta

Probability and expected frequency · Revision guide

Foundation · Probability · 1 page in the PDF

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Learning objectives

  • Calculate probabilities and use complementary outcomes.
  • Find an expected frequency and combine independent events.

Key ideas

For equally likely outcomes, count favourable outcomes out of the total.

P(A)=favourable outcomestotal outcomesP(A)=\frac{\text{favourable outcomes}}{\text{total outcomes}}

Probabilities lie between zero and one; all mutually exclusive exhaustive outcomes total one.

The probability of an event not happening is its complement.

P(not A)=1−P(A)P(\text{not }A)=1-P(A)

An expected frequency predicts a long-run average, not a guaranteed result.

E=number of trials×P(A)E=\text{number of trials}\times P(A)

Multiply probabilities to combine independent events; without replacement, probabilities may change.

P(A and B)=P(A)P(B)if independentP(A\text{ and }B)=P(A)P(B)\quad\text{if independent}

Worked examples

Example 1

Predict the number of even results in ninety rolls of a fair six-sided die.

even outcomes: 2,4,6P(even)=36=12\text{even outcomes: }2,4,6\qquad P(\text{even})=\frac36=\frac12
E=90×12=45E=90\times\frac12=45
45 even results expected45\ \text{even results expected}

Example 2

A fair coin and fair six-sided die are independent. Find the probability of heads and a six.

P(heads and six)=12×16=112P(\text{heads and six})=\frac12\times\frac16=\frac1{12}
112\frac1{12}

Watch out

Expected frequency is not a promise. Multiplication of unchanged probabilities needs independent events.