The Probability Scale

From impossible to certain

Some things are sure to happen, some can never happen, and most are somewhere in between. Something that might happen, such as rolling a 6, is called an event.

Rolling a 7 on a dice numbered 1 to 6 can never happen: it is impossible. Rolling a number less than 7 happens on every roll: it is certain.

A probability is a number that says how likely an event is. An impossible event has probability 0, and a certain event has probability 1. Nothing is less likely than impossible, and nothing is more likely than certain, so every probability is a number from 0 to 1.

The probability scale runs from 0, impossible, to 1, certain. Every event has its place somewhere on this line.

Halfway is an even chance

A fair coin lands heads or tails, and the two are equally likely. Heads is as likely to happen as not to happen, so its probability sits halfway along the scale, at $\frac{1}{2}$1/2. A probability of $\frac{1}{2}$1/2 is called an even chance.

Events between 0 and $\frac{1}{2}$1/2 are unlikely: they happen less often than not. Events between $\frac{1}{2}$1/2 and 1 are likely: they happen more often than not.

The colored mark is an even chance, exactly $\frac{1}{2}$1/2. Unlikely events lie between 0 and $\frac{1}{2}$1/2, and likely events lie between $\frac{1}{2}$1/2 and 1.

Counting equal chances

A spinner has 6 equal sections, numbered 1 to 6. The sections are all the same size, so the pointer is equally likely to stop on any of them. Only one section shows a 3. That is one win in six, so the probability of spinning a 3 is $\frac{1}{6}$1/6.

$\frac{1}{6}$1/6 is much less than $\frac{1}{2}$1/2, so spinning a 3 is unlikely. It sits near the impossible end of the scale, but not on it: a 3 can happen, just not often.

One of the 6 equal sections shows a 3, so the probability of spinning a 3 is $\frac{1}{6}$1/6.

$\frac{1}{6}$1/6 is close to 0, a long way short of the even chance at $\frac{1}{2}$1/2.

The same spinner, from end to end

The pointer can never stop on a 7, because no section shows a 7. That is 0 wins in 6, and $\frac{0}{6}=0$0/6 = 0: impossible. The pointer always stops on a number less than 7, because every section shows one. That is 6 wins in 6, and $\frac{6}{6}=1$6/6 = 1: certain.

Three of the six sections show an even number: 2, 4 and 6. That is 3 wins in 6, and $\frac{3}{6}=\frac{1}{2}$3/6 = 1/2, an even chance, the same as heads on a coin.

3 of the 6 equal sections show an even number: $\frac{3}{6}=\frac{1}{2}$3/6 = 1/2, an even chance.

Never below 0, never above 1

A probability can never be more than 1, because nothing is more likely than certain, and it can never be less than 0, because nothing is less likely than impossible. So if a working gives a probability of $\frac{6}{1}$6/1 or $\frac{3}{2}$3/2, there is a slip in it: the number of wins has to go on top, and the number of chances in all on the bottom.