Chance and the Probability Scale

You already talk about chance every day. You say something is certain, or likely, or impossible, without thinking about it. A dropped ball falling to the ground is certain. Rolling a number less than 5 on a die is likely, because four of the six faces work. A tossed coin landing on heads is an even chance, because heads and tails are just as good as each other. Rolling a 6 is unlikely, since only one face out of six gives it. Rolling a 9 on an ordinary die is impossible. Those five words run in order from never to always, and in this chapter you will turn each of them into a number so you can compare chances exactly.

Here is the picture that holds the whole topic together. Probability is measured on a scale that runs from 0 to 1. Zero sits at the far left and means the event cannot happen. One sits at the far right and means it must happen. Halfway along, at one half, is an even chance, where the event is just as likely to happen as not. A quarter of the way along is unlikely, and three quarters of the way along is likely. Nothing lives outside this scale. No chance is smaller than 0, and no chance is bigger than 1, so if you ever calculate something like 2, or a negative number, you know straight away you have made a mistake.

Now let us put real events on the scale. Rolling a 9 on an ordinary die is impossible, so it sits at 0, right at the left end. Rolling a 6 has one chance in six, so it sits close to the left but not on it. A tossed coin landing on heads is one half, so it sits exactly in the middle. Rolling a number bigger than 2 means a 3, 4, 5 or 6, which is four faces out of six, or two thirds, so it sits to the right of the middle. And the sun rising tomorrow is certain, so it sits at 1. The further right a dot sits, the more likely that event is.

A probability can be written three ways, and they all mean the same thing. Look at this spinner with four equal sections: two red, one blue and one green. Landing on red is two sections out of four, which is one half, or 0.5, or 50 percent. Landing on blue is one section out of four, which is a quarter, or 0.25, or 25 percent, and green is the same. Notice that the three probabilities add up to 1, or to 100 percent, because the arrow has to land somewhere. Fractions are usually the safest to work with because nothing gets rounded, but you should be comfortable swapping between all three forms.

Finally, how do you decide which of two chances is bigger? Bag A holds four counters and one of them is red, so the probability of red is one quarter, or 0.25. Bag B holds five counters and three of them are red, so the probability of red is three fifths, or 0.6. Comparing one quarter with three fifths in your head is awkward, but comparing 0.25 with 0.6 is easy. That is the trick: put both chances into the same form first, usually decimals or percentages, then compare. On the scale, Bag B's dot sits further to the right, so Bag B gives you the better chance of pulling out a red counter.