C8.1; E8.1 - Single-event probability

Probability scale and equally likely outcomes

A bag contains three red counters and one blue counter. Is blue just as likely as red because there are two colours? Imagine choosing one counter with each individual counter equally likely. Three of the four possible counters are red: counting colour names would miss the different numbers of counters.

For the bag, red has probability $\frac34$ and blue $\frac14$. A probability may also be a decimal or percentage: $\frac34=0.75=75\%$. These describe the same chance, not three different predictions.

[DIAGRAM: asset_name: The probability scale; asset_slug: probability_scale; caption: Probabilities run from zero to one, with an even chance at one half.; description: A horizontal evenly spaced scale from 0 to 1 has marked positions 0, one quarter, one half, three quarters and 1. The endpoints are labelled impossible and certain, the centre even chance, and the quarter and three-quarter positions less likely and more likely than not.]

An event with probability $\frac12$ has an even chance. A probability below $\frac12$ is less likely to happen than not; one above $\frac12$ is more likely to happen than not. Values such as $-0.2$ and $1.3$ cannot be probabilities.

Keep the die, but change only the event. One event can contain several outcomes.

The equal-chance condition matters. On a biased die, the six face labels still exist, but dividing by six need not give their probabilities. You would need information about their actual chances. Random means the particular result is uncertain; it does not by itself make a chosen list of categories equally likely.

For the die below, the equal-chance condition is supplied; identify the favourable faces.

Counting a single event

When choosing a person at random from a group, make the intended model explicit: every person is equally likely to be chosen. Then a table, bar chart or Venn diagram can supply the favourable count and the whole-group count. Read what the numbers represent before calculating.

The chart shows the travel methods of $20$ students. Each student is in exactly one category, so the bar heights add to the total.

[DIAGRAM: asset_name: Travel-method counts; asset_slug: probability_travel_counts; caption: Use the frequency scale to count students, rather than counting the three bars.; description: A bar chart has categories Walk, Bus and Cycle with frequencies 8, 7 and 5. The vertical frequency axis starts at zero, is marked in steps of one and reaches ten. Each bar has its frequency printed above it.]