Calculating the probability of an event as a fraction, decimal or percentage, interpreting probability on the scale from 0 to 1, using expected frequency, and using probability and risk to make and justify decisions.

What this dot point is asking

The SQA wants you to calculate the probability of an event as a fraction, decimal or percentage, place it on the likelihood scale from $0$ to $1$, work out an expected frequency, and use probability and risk to make and justify a decision.

Calculating probability

Probability measures the chance of an event happening. When all outcomes are equally likely, it is a simple fraction of favourable outcomes over the total.

The likelihood scale and complementary events

Every probability sits on a scale from $0$ to $1$. An impossible event has probability $0,$ a certain event $1$, and an even chance $0.5$.

Expected frequency, risk and decisions

Expected frequency predicts how many times an event should happen over many trials. It turns a probability into a count, which helps weigh up risk.

Risk questions ask you to use a probability to judge how safe or worthwhile something is, and to justify the decision. A low probability of a serious fault may still matter if the consequences are severe, so the decision must be explained, not just stated.

Probability can be found from data as well as from equally likely outcomes. If a bus was late on $15$ of the last $60$ days, the experimental probability of it being late is $\dfrac{15}{60} = \dfrac{1}{4}$, which can then be used to predict future lateness. The more data you have, the more reliable this estimate becomes, which is why expected frequency over many trials is trusted more than a single result.

Comparing two options on risk uses the same idea: the option with the lower probability of a bad outcome, or the lower expected number of failures, is usually the safer choice, but the size of the consequence matters too.

Examples in context

Probability and risk underpin everyday choices: a weather forecast gives the chance of rain, a manufacturer estimates how many products will be faulty, an insurer prices a policy on the risk of a claim. The SQA tests calculating the probability, finding an expected frequency, and using these to make and justify a decision, the skills here.

Try this

Q1. A fair coin is tossed. Find the probability of heads. [1 mark]

• Cue. $\dfrac{1}{2}$.

Q2. $P(\text{late}) = 0.15$. Find the probability of not being late. [2 marks]

• Cue. $1 - 0.15 = 0.85$.

Q3. $P(\text{prize}) = 0.1$ and $200$ tickets are bought. Find the expected number of prizes. [2 marks]

• Cue. $0.1 \times 200 = 20$.