11.01a, 11.01d - Probability and expected outcomes

The probability scale

A standard six-sided dice can show a six, but it cannot show a seven. Both are descriptions of an event, something we are interested in happening. An outcome is one possible result of an experiment, such as a particular face showing.

On a fair standard dice, showing seven is impossible, showing a number from one to six is certain, and showing an even number has an even chance. An event below $0.5$ is less likely than not; one above $0.5$ is more likely than not. Probability can be written as a fraction or decimal: $1/4=0.25$, so these are the same position on the scale.

[DIAGRAM: asset_name: The probability scale; asset_slug: probability-scale; caption: Probability runs from zero for impossible to one for certain.; description: A horizontal probability scale marks zero impossible, one quarter unlikely, one half evens, three quarters likely and one certain. Fractions and decimals are paired at the internal marks.]

A small positive probability is still possible. An event with probability $0.01$ may happen on the very next trial, while an event with probability $0.99$ may fail to happen. “Likely” is not another word for “certain”.

Choose the statement that keeps chance separate from certainty.

Counting equally likely outcomes

Imagine ten identical counters in a bag, four blue and six red. Mix them and choose one without looking. Each individual counter has the same chance of being selected, so count counters, not just colour names.

The formula requires equal likelihood. With four blue and six red counters, the two colours are not equally likely merely because there are two colour names. Choosing a counter at random and hiding its colour makes the individual counters a reasonable equally likely model.

Change only one counter from red to blue. The total stays ten, while the favourable count increases by one.

For a fair dice, the equally likely outcomes are its faces. For a fair spinner divided into equal sectors, they are its sectors. Repeated numbers on separate sectors still count separately. A spinner with two sectors labelled four has two ways to show four.

Use the favourable count in a different bag.

Randomness and fairness

Random describes uncertainty in an individual result. Fair describes the chances in the model or the rules of a game. They are connected, but they are not interchangeable.

A fair coin has equal probabilities of heads and tails. A fair dice has equal probabilities for its faces. A selection is random when it uses chance rather than choosing a particular result; it can still give unequal probabilities to the categories we care about. A random counter from a bag containing nine red and one blue is much more likely to be red.

For two players with equal rewards and equal stakes, a fair game gives them equal chances of winning. A fair dice does not automatically make every game using it fair. Suppose one player wins on a six and the other wins on any other face. The chances are $1/6$ and $5/6$, so the rules favour the second player despite the fair dice.

Outcomes can also be independent: knowing one result does not change the probabilities for the other. For independent tosses of a fair coin, heads on the previous toss leaves the next heads probability at $1/2$. The coin has no need to “catch up” after a run of tails. This assumption is about the experiment; it is not automatic for every sequence of random events.