Probability in Decision-Making
Explore probability in decision-making for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). News outlets, governments and businesses use statistics and probabilities to persuade people and to make decisions, so it pays to check their claims yourself.
You will look at how statistics and the chances behind them shape decisions, test claims using probability, expected frequency and relative frequency, and spot misleading samples and graphs — skills for judging raffles, advertisements, surveys and weather forecasts in everyday life.
Theory
Probability in decision-making means using probability to test the claims made by the media, governments and companies. This Year 12 Standard 1 (NSW) guide shows how to check raffles, advertisements and surveys with probability, expected numbers and relative frequency, and how to spot misleading samples and graphs.
The media, governments and companies use statistics and probabilities to persuade people and to make decisions. Check every claim: work out the probability or the expected number yourself, and ask how the data was collected and presented.
Use the tools you already have: \(P(\text{event})=\dfrac{\text{favourable}}{\text{total}}\), \(P(\text{not }A)=1-P(A)\), expected number \(=n\times p\), relative frequency as an estimate of \(p\), and listing the equally likely outcomes of two stages in a tree diagram or array. A forecast such as “\(70\%\) chance of rain” is a long-run rate: on about \(70\) of every \(100\) days like this, it rained. It is not a promise.
Ask about any statistic: how big was the sample and how was it chosen? Is it an association being sold as causation? Does “doubled” come from a small base? Is the graph’s axis cut off, data left out (cherry-picking only the best months), or a mean pulled up by a few very large values?
On a scatter plot an association shows as points that trend up or down; a shapeless cloud of points shows no association.
To test a claim, calculate the numbers yourself:
Overlapping groups. Count people in both groups only once: \(60+40-16=84\) pet owners when \(60\) own a dog, \(40\) a cat and \(16\) both.
How to check a claim
- Calculate the probability, relative frequency or expected number from the figures given.
- Compare your answer with the words of the claim (“always”, “probably”, “a great chance”).
- Question the data: sample size and how it was chosen, association versus causation, small bases, and how any graph is drawn.
Each ticket is equally likely to be drawn.
| \(P(\text{win})\) | \(=\) | \(\dfrac{5}{400}=0.0125\) |
| \(\) | \(=\) | \(0.0125\times100\%=1.25\%\) |
| \(P(\text{not win})\) | \(=\) | \(1-0.0125=0.9875\) |
\(P(\text{win})=0.0125\) (\(1.25\%\)). Not fair — you are almost certain not to win.
Use the relative frequency as \(p\), then expected \(=n\times p\).
| \(\text{expected missing}\) | \(=\) | \(25\,000\times0.004=100\) |
| \(\text{expected to arrive}\) | \(=\) | \(25\,000-100=24\,900\) |
About \(100\) bags are expected to go missing, so “always” is not a fair claim.
Find the relative frequency, then question the sample.
| \(\text{(i) RF}\) | \(=\) | \(\dfrac{10}{12}\approx0.83=83\%\) |
(ii) Only \(12\) people is a very small sample, and they were all gym members who chose the program — not a fair picture of “people”.
(iii) Survey a large, randomly chosen group of people.
(i) \(0.83\) — but the claim is not reliable.
Label the sectors \(W, L_1, L_2, L_3, L_4\). Two spins give \(5\times5=25\) equally likely outcomes; the shaded cells contain a win. No win: \(4\times4=16\) cells.
| \(P(\text{no win})\) | \(=\) | \(\dfrac{16}{25}\) |
| \(P(\text{at least one win})\) | \(=\) | \(1-\dfrac{16}{25}=\dfrac{9}{25}=0.36\) |
\(P=0.36\). This is less than \(0.5\), so the claim is false: no win (\(0.64\)) is more likely.
Common pitfalls
Frequently asked questions
How is probability used in decision making?
Probability and expected numbers let you test claims and compare choices. For example, an airline that loses 0.004 of bags should expect about 100 missing bags in 25 000, so it cannot claim bags always arrive.
What does a 70% chance of rain mean?
It means that on about 70 out of every 100 days with conditions like this, it rained. It is a long-run rate, not a promise that it will rain.
How can statistics be misleading?
Statistics can mislead through small or hand-picked samples, graphs with the vertical axis cut off, data left out (cherry-picking, such as showing only the best months), means pulled up by a few large values, and percentages of very small bases.
Why does sample size matter in a survey?
A small sample can give a result far from the truth just by chance, and a hand-picked sample may not represent everyone. A large, randomly chosen sample gives a more reliable estimate.
What is the difference between association and causation?
An association means two things change together. Causation means one actually causes the other. Two things can rise together because a third factor drives both, so an association alone does not prove cause.
What does probably mean in probability?
Probably means the probability is greater than 0.5, so the event is more likely to happen than not. Spinning twice a wheel with 1 winning sector out of 5 gives 25 equally likely outcomes and only 9 include a win, so the chance is 0.36 and a win is not probable.