Probability, relative frequency and expected outcomes: QCE Essential Mathematics Unit 3
“Describe the likelihood of events using the language of chance and the probability scale from 0 to 1, list sample spaces, calculate theoretical probabilities, and use relative frequencies from experiments and data to estimate probabilities and expected numbers of outcomes”
Probability runs from 0 (impossible) to 1 (certain). For equally likely outcomes, divide favourable outcomes by total outcomes. Estimate probabilities from data with relative frequency (occurrences divided by trials), which becomes more reliable with more trials, and find expected outcomes by multiplying probability by the number of trials.
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What this dot point is asking
Unit 3's title includes "chance". You need to describe how likely events are, calculate probabilities when outcomes are equally likely, use relative frequency from experiments or data to estimate probabilities, and predict how many times an outcome is expected. Our heading here follows the unit title; check your school's unit plan for the exact subject matter.
The answer
The language and scale of chance
Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
| Words | Probability |
|---|---|
| Impossible | 0 |
| Unlikely | between 0 and 0.5 |
| Even chance | 0.5 |
| Likely | between 0.5 and 1 |
| Certain | 1 |
Sample spaces and theoretical probability
The sample space is the list of all possible outcomes. When outcomes are equally likely:
The probability an event does not happen is .
Tables and tree diagrams help list outcomes for two steps (such as two coins or a coin and a die).
Relative frequency
When outcomes are not equally likely, or we do not know the probability, we estimate it from experiments or data:
Relative frequency from a small number of trials can vary a lot. As the number of trials increases, relative frequency gets closer to the true probability.
Expected number of outcomes
Enterprises use this to plan: expected sales of each item, expected faulty products, expected insurance claims.
Over the last 5 years, it rained on 52 of the 150 days in December, January and February at a local showground.
- Relative frequency of rain = 52/150 = 0.347 (about 35%).
- A 3-day summer festival is planned. Expected rainy days = 0.347 × 3 = 1.04, so about 1 day.
- Decision: the organisers should hire marquees, since rain on at least one day is likely.
- Limitation: weather varies from year to year, so this is an estimate, not a forecast.
- Probabilities above 1 or below 0
- They are impossible; check your working.
- Confusing relative frequency with probability
- Relative frequency is an estimate from data.
- Expecting exact results
- Expected numbers are averages over many trials, so actual results vary.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksA bag contains 5 red, 3 blue and 2 green counters. One counter is drawn at random. Find the probability that it is (a) red, (b) not blue, (c) yellow.Show worked solution →
Total = 10 counters.
(a) P(red) = 5/10 = 0.5.
(b) P(not blue) = 7/10 = 0.7 (or 1 minus 3/10).
(c) P(yellow) = 0/10 = 0 (impossible).
Marking guide: 1 mark each.
core4 marksA café records the drinks ordered by its last 200 customers: 84 flat whites, 46 cappuccinos, 38 lattes and 32 other drinks. (a) Find the relative frequency of a flat white. (b) The café expects 350 customers on Saturday. How many flat whites should it expect to make? (c) Why is this only an estimate?Show worked solution →
(a) Relative frequency = 84/200 = 0.42.
(b) Expected flat whites = 0.42 × 350 = 147.
(c) The relative frequency comes from a sample of past customers. Saturday customers may have different preferences (more families, different weather), and there is natural variation from day to day, so the actual number will differ.
Marking guide: 1 mark for (a), 1 mark for (b), 2 marks for (c) with a reason.
exam5 marksA factory tests light globes. In a sample of 400 globes, 14 were faulty. The factory produces 12 000 globes a week and replaces faulty globes free of charge at a cost of 3 dollars each. (a) Estimate the probability that a globe is faulty. (b) Estimate the weekly cost of replacements. (c) Evaluate how reliable your estimate is and suggest how to improve it.Show worked solution →
(a) P(faulty) is estimated by the relative frequency 14/400 = 0.035 (3.5%).
(b) Expected faulty globes = 0.035 × 12 000 = 420. Cost = 420 × $3 = $1260 per week.
(c) A sample of 400 is reasonably large, so the estimate is useful, but it is from one sample. A different sample could give, say, 10 or 18 faulty globes, changing the cost estimate. The factory should test more globes, sample from different shifts and machines, and repeat over several weeks. Relative frequency becomes a more reliable estimate of probability as the number of trials increases.
Marking guide: 1 mark for (a), 2 marks for (b), 2 marks for (c) including why more trials help.