Calculating Probabilities

Now you can turn a sample space into a number. The probability of an event is the number of ways it can happen, divided by the total number of outcomes. Take a fair six-sided die. There are six outcomes altogether, so six always goes on the bottom. Rolling a 3 can happen in one way, so that is one out of six. Rolling an even number can happen three ways, since 2, 4 and 6 are all even, so that is three out of six, which simplifies to one half. Rolling more than 4 can only happen with a 5 or a 6, so that is two out of six, or one third.

Bag questions work exactly the same way, but you have to find the total yourself. This bag holds 3 red, 2 blue and 1 yellow counter, so the total is 3 plus 2 plus 1, which is 6. That 6 goes on the bottom of every fraction. The probability of red is 3 out of 6, which simplifies to one half. Blue is 2 out of 6, which is one third. Yellow is 1 out of 6. And green is 0 out of 6, which is just 0, because there is no green counter in the bag at all. The mistake to avoid is putting 3 on the bottom when you are asked about red. The bottom is always the total.

Once you have your fraction, finish the job by simplifying it. To simplify, divide the top and the bottom by the same number. Three sixths becomes one half when you divide both by 3. Two sixths becomes one third when you divide both by 2. Four eighths becomes one half. Two tenths becomes one fifth. Writing three sixths is not wrong, but one half is the finished answer and that is what you should hand in. You can also give a probability as a decimal or a percentage, so one half is 0.5 or 50 percent. But be careful with thirds, because one third as a decimal is 0.333 repeating and never ends, so the fraction is safer.

The same bag can be asked about in different ways, and the wording tells you what to count. The word 'or' means add the groups together, so the probability of red or yellow is 3 plus 1, which is 4 out of 6, or two thirds. The word 'not' means count everything else, so the probability of not red is the 2 blue plus the 1 yellow, giving 3 out of 6, or one half. Blue or yellow is 2 plus 1, again 3 out of 6. And red, blue or yellow covers every counter, so that is 6 out of 6, which equals 1 — certain, because the counter you pull out must be one of those three colours.

Let us finish with the routine you should use every time. First, find the total number of outcomes and write it down rather than keeping it in your head. Second, count the ways your event can happen, watching for words like 'or' and 'not'. Third, write the fraction with that count on top and the total underneath. Fourth, simplify it and check the answer sits between 0 and 1. One more habit worth building: use the P bracket notation, so you write P of red equals one half rather than just one half. It shows the marker which event you found, and that matters when a question asks about several different events in a row.