Multi-step probability

How likely is it that a coin shows heads twice in a row? Or that you draw a red marble from a bag twice in a row? As soon as chance has several steps, gut feeling won't get you far. Luckily there is a rule that makes everything clear.

The path rule

Draw a tree diagram and write on each branch the probability of taking it. For an outcome across several steps, follow its path and multiply the probabilities along the way. Heads twice with a fair coin: each flip shows heads with probability 0.5, so together 0.5 · 0.5 = 0.25, which is one quarter. That is the path rule, and it works for every multi-step random experiment.

With or without replacement

With marbles it matters whether you put the drawn marble back. Bag with 2 red and 3 blue marbles: with replacement, the chance of red is 2 out of 5 on every draw. Without replacement, the bag changes after the first draw: after a red marble only 1 red and 3 blue remain, so the chance of red drops to 1 out of 4. The path rule still applies, you just write the new probabilities on the branches.

The complement trick

Some questions have many matching paths, for example: at least one heads in two flips. Instead of collecting all those paths, use the complement: no heads at all. That is just one path: 0.5 · 0.5 = 0.25. So the probability of at least one heads is 1 − 0.25 = 0.75. For questions with at least, this detour is almost always worth it.