Tree diagrams make the sample space of a two-step experiment visible

0:00Everything so far has been one step. Roll a die once. Draw one ball. Toss one coin. Now do two things in a row, and the experiment is called multi-step. Toss a coin twice. Draw a ball, then another. Take a fruit from one basket and one from a second. The list of outcomes gets longer, and writing it out by hand starts going wrong. So there is a picture for it, called a tree diagram, and it is the subject of this video.

0:33It looks like an organiser. It is not: it is two multiplications drawn. Here is the tree for a fair coin tossed twice, with four things worth naming. From a starting point, two lines go out. Each line is a branch, and each branch stands for one result of one step. The first toss gives head or tail, so two branches. From the end of each, two more branches, because the second toss also gives head or tail.

1:05A complete route from the start to an end is a path, and the whole tree has six lines drawn on it. Read off the ends and you get head-head, head-tail, tail-head, tail-tail. That is your sample space. The rest of this video is about the numbers on those six lines. First, why four ends and not three or five. There were two ways the first step could go, and from each of those, two ways the second could go.

1:36So the count is not two plus two. It is two times two. That is the first multiplication the tree draws, and it does not stop at two steps. Roll a die three times and the tree has six branches, then six from each of those, then six again. Six times six times six is two hundred and sixteen paths, and nobody is drawing that. You do not need to. The branchings multiply, and the number comes out without the picture.

2:10Now the numbers. Every one of the six branches on the coin tree carries a half. It is worth being exact about what that half is the probability of, because it is easy to get wrong. It is the probability of that one step going that one way. Not the probability of the whole outcome. Look at the second layer. The branch marked head means: given the first toss already happened, this one comes up heads.

2:39A half. Every time, on every branch, on this particular tree. And notice what is not written anywhere on the picture: a quarter. No branch carries the probability of a complete result. So how do you get the probability of two heads? One way is to ignore the numbers entirely and just count. Four paths, one of them is head-head, so one quarter. That is the method from earlier: favourable outcomes over possible outcomes.

3:13Try it on something with more than one path. What is the probability of one head and one tail? Two of the four paths qualify, head-tail and tail-head, so two over four, a half. Notice what happened there. The answer came from two separate paths, and they had to be put together. Hold on to that, because it is the second thing a tree makes you do, and it is not multiplying.

3:42Here is the other way, and it is the one the picture is asking for. Follow the top path. Half on the first branch, half on the second. Multiply them. A half times a half is a quarter, the same answer counting gave. That is the second multiplication the tree draws: along a path, the numbers multiply. Be careful: at a single split those same numbers add, because something has to happen. Adding two halves gives one; multiplying them gives a quarter.

4:17Down a path, multiply. Across a split they should already add to one, which is your check that you drew it right. For one head and one tail, each path is a quarter and the two quarters add to a half. Two methods, same answer. So why bother with the multiplying? Because now change one thing. One basket holds an apple and two oranges. A second holds a banana and a mango.

4:47Take one fruit from each. What is the probability of an apple with a banana? The tree still has four ends: apple-banana, apple-mango, orange-banana, orange-mango. Count them and you get one in four. That answer is wrong. The first two branches are not halves. Apple is one of three fruits, a third, and orange is two of three, two thirds. Multiply along the apple-banana path: a third times a half is a sixth.

5:20A sixth, not a quarter, and the gap is a twelfth. The tree was right and the counting was not. Counting is not a broken method, so it is worth seeing what went wrong. There are really six ways to pick, because the two oranges are two different oranges, and all six are equally likely. Count in that list of six and the sixth comes out correctly. The four-outcome list is those six with the oranges no longer told apart, and merging is not free.

5:55Apple with banana is one of the six. Apple with mango is one. Orange with banana is two, and orange with mango is two. So the four ends carry a sixth, a sixth, a third and a third. Not a quarter each, and never were. Counting paths works only when the paths weigh the same, which on a fair coin they do and here they do not. It gets worse, and the next example shows how much.

6:26A box holds three red pens, four black and two green. Take one, look, put it back, and take another. The first layer now has three branches: three ninths, four ninths, two ninths. What is the probability that both pens are the same colour? Nine colour-paths, three of them same-colour, so counting says a third. Multiply instead. Red then red is nine eighty-firsts. Black then black is sixteen. Green then green is four.

6:58Add those three, because any of them will do, and you get twenty-nine eighty-firsts. A third is twenty-seven eighty-firsts, so counting is short by two, and this time it is not obvious at all. One more, and this one changes the tree itself. A basket holds four red balls and five blue. Draw one, keep it out, and draw again. The first layer is four ninths and five ninths. But the second layer depends on what you drew, because one ball is gone and eight are left.

7:35Down the red branch three reds remain out of eight, so three eighths and five eighths. Down the blue branch four reds remain, so four eighths and four eighths. Red then blue is four ninths times five eighths, five eighteenths. Two blues is five ninths times four eighths, also five eighteenths. The same number, and the picture shows why: the same four numbers, in the other order. Do not read that as a rule. Four red and four blue, and they come apart at two sevenths and three fourteenths.

8:13That last tree is worth a moment, because it breaks a definition. A multi-step experiment is usually defined as a series of independent trials: what happens first does not change the odds of what happens next. The coin is like that. The second toss does not know or care what the first one did. The balls are not. Taking a red one out changes the second layer, and the tree shows exactly how.

8:44So the definition is too narrow for its own examples. The drawing does not care. Nothing about the picture changed: same shape, same lines, different numbers on the second layer. Independence is a property of the experiment, and the tree just reports it. Two last things. First, you do not always want a tree. Two shirts and three kinds of trousers give six outfits, and a two-by-three table shows them better.

9:16Use a table for two steps with several choices each, and a tree when the branch numbers differ or there are three steps or more. Second, for a size you need no picture at all. Multiply the branchings. Four numbered balls drawn twice, putting the first back: four times four is sixteen. Not putting it back: four times three is twelve. A coin with one of six cards is two times six, twelve.

9:46So: the ends multiply, the path multiplies, the split adds, and count the ends only when they weigh the same. Four sentences, and the picture holds all of them at once.