1.7 Reading information
Probability
Everything on probability that can come up in the WJEC GCSE Mathematics and Numeracy, in all three units. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
A worked example
Probability · [1 mark]
A bag contains 5 red, 6 blue and 1 green counters. One counter is taken at random. What is the probability that it is not red?
- B17/12
The exam skills (11)
Probability
- Count the outcomes you want.
- Count all the outcomes.
- Probability = wanted ÷ total, as a fraction.
Watch out: Writing the answer as a ratio like 1 : 2. Probabilities are fractions, decimals or percentages.
Expected outcomes
- Expected number = probability × number of trials.
- Multiply.
- It is what you would expect on average, so it can be a decimal.
Watch out: Dividing by the probability instead of multiplying.
Relative frequency
- Relative frequency = number of times it happened ÷ number of trials.
- Use it as an estimate of the probability.
- Expected number = estimate × new number of trials.
Watch out: Dividing by the number of times it did not happen.
Sample spaces
- Draw a sample space: 6 × 6 = 36 outcomes.
- Count the outcomes that fit.
- Probability = count ÷ 36, then simplify.
Watch out: Thinking there are 12 outcomes because there are two dice.
Tree diagrams
- Draw a tree diagram. Put it back means the probabilities stay the same.
- Multiply along the branches for "and".
- Add the results of different routes for "or".
Watch out: Adding along the branches. Along = multiply.
Venn diagrams
- Fill in the overlap (both) first.
- Then fill in the rest of each circle, then the outside.
- Probability = how many in the region you want ÷ total.
Watch out: Counting the overlap twice.
Mutually exclusive events
- Mutually exclusive means they cannot happen at the same time.
- P(A or B) = P(A) + P(B).
- P(neither) = 1 − P(A or B).
Watch out: Multiplying. That is for independent events happening together.
Without replacement
- After the first pick there is one fewer in the bag.
- So the second fraction has a bottom one smaller, and the top may change too.
- Multiply along the branches.
Watch out: Keeping the same bottom number for the second pick.
At least one
- At least one = 1 − P(none at all).
- P(none) = P(not) multiplied once for each go.
- Take it away from 1.
Watch out: Listing every way to get at least one. It is long and easy to miss one. Use 1 − P(none).
Two-way tables
- Read the question: "a Year 11 student is picked" means only use that row or column.
- The total of that row or column is your bottom number.
- The cell you want is your top number.
Watch out: Dividing by the grand total when the question says "given" or picks from one group.
Three events
- Write one route as three probabilities multiplied, e.g. late × on time × on time.
- Count the routes that give the same result (usually 3).
- Multiply one route by the number of routes.
Watch out: Only counting one route.
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