Sample Spaces and Equally Likely Outcomes

Before you can work out any probability, you need to know what could happen. One thing that could happen is called an outcome, and the full list of all the outcomes is called the sample space. Toss a coin and there are two outcomes, heads and tails, so the sample space is heads, tails. Roll an ordinary die and there are six outcomes, so the sample space is 1, 2, 3, 4, 5, 6. We usually write the list inside curly brackets, and each outcome goes in exactly once. That list is the foundation for everything else in this chapter, so get into the habit of writing it down before you try to calculate anything.

The skill here is listing outcomes without missing any, and the trick is to work in a fixed order rather than jumping about. Picking a day of the week at random gives seven outcomes: Monday through to Sunday, in order. Two of those, Saturday and Sunday, are the weekend. Now picking one letter from the word BANANA. There are six letters, so there are six outcomes, even though only three different letters appear. That catches people out. You are picking one letter tile, not choosing between the names B, A and N, so all six tiles count. Go through in order and count as you go, and you will not miss anything.

Here is an idea you must not skip over. Outcomes are equally likely when each one has just as much chance as the others. The spinner on the left has four equal sections, so landing on 1, 2, 3 or 4 are equally likely, and each has a one in four chance. The spinner on the right also has sections, but they are not the same size. Red covers half the circle, while blue and green each cover a quarter. So even though there are three outcomes, they are not equally likely. Red has a one half chance, not one third. Always check the sizes before assuming. Fair dice, fair coins and evenly divided spinners give equally likely outcomes.

Now for two things happening at once. Toss two coins and you can draw a tree. The first coin is either heads or tails. Whichever it is, the second coin is also either heads or tails. That gives four branches at the end: HH, HT, TH and TT. So the sample space has four equally likely outcomes, which makes sense because two sides times two sides is four. Here is the trap. People often write only three outcomes, thinking two heads, two tails, and one of each. But one of each happens in two different ways, head then tail, and tail then head, so both belong in the list.

When two different things happen together, a table is the neatest way to lay out the sample space. Put the coin down the side, heads and tails, and the die across the top, 1 to 6. Every cell in the table is one outcome, like heads with a 3, or tails with a 5. Count the cells and you get two rows times six columns, which is twelve outcomes altogether. All twelve are equally likely, so each has a one in twelve chance. Watch out for a common slip here: some people add 2 and 6 to get 8. You multiply, because every one of the two coin results pairs up with every one of the six die results.