C9.1; E9.1 - Probability and complements

Probability scale

If every counter is red, choosing a red counter is certain. If no counter is red, it is impossible. Between these extremes, probability measures how likely an event is.

On a fair six-sided die, “roll 4” is one outcome. “Roll an even number” is an event containing 2, 4 and 6. Half the equally likely outcomes are even, so its probability is $\frac12$.

[DIAGRAM: asset_name: Probability scale; asset_slug: probability_scale; caption: Zero is impossible, one half is an even chance and one is certain.; description: A horizontal scale from zero to one, with ticks at zero, one quarter, one half, three quarters and one. Labels below show impossible at zero, unlikely below one half, even chance at one half, likely above one half and certain at one.]

The equivalent forms $\frac12=0.5=50\%$ all mean an even chance. A probability below $\frac12$ is less likely than not; a probability above $\frac12$ is more likely than not. “Unlikely” still allows the event to happen.

For a bag with 9 red counters and 1 blue counter, choosing red has probability $\frac9{10}=0.9$. It is likely, but not certain: the blue counter remains possible. Neither $-0.2$ nor $1.2$ can be a probability.

Read the boundary values carefully.

Single events

Counting gives a probability when the individual objects or outcomes are equally likely. First identify everything that could be selected, then count the outcomes satisfying the event.

“At random” in a selection of counters means each individual counter has the same chance. Colours need not have the same chance: five red counters provide five favourable objects, not just one colour name.

A table can collect repeated objects. Add the frequencies, rather than counting the number of table headings.

A graph displays the same kind of counts. Read the height against the labelled scale; a bar's colour or width does not give its frequency.

[DIAGRAM: asset_name: Stickers of four shapes; asset_slug: sticker_counts; caption: The bar heights count 18 stickers altogether.; description: A bar chart has horizontal categories circle, triangle, square and star and a vertical number-of-stickers scale from zero to eight in unit steps. The four bar heights are four, seven, five and two. Each height is also labelled.]