Mapping Every Possibility: Tree Diagrams

Imagine you flip a coin and then spin a spinner with three equal sections: Red, Blue, and Green. How many different results could you get in total? Could you list them all without missing any? Most people try to list outcomes in their head and accidentally miss some — especially when the number of choices grows. There must be a smarter, more reliable method. Today you will discover exactly that.

A tree diagram is a branching picture that shows every possible outcome of a series of events. Each branch represents one possible result at each stage. By following a path from left to right through the branches, you read off one combined outcome. Here is how to build one for flipping a coin (Heads or Tails) and then spinning a Red/Blue/Green spinner. STEP 1 — Draw the first set of branches. From a starting point on the left, draw one branch up labelled H (Heads) and one branch down labelled T (Tails). These are the two outcomes for the coin flip. STEP 2 — Draw the second set of branches. From the end of EACH coin branch, draw three new branches labelled R, B, and G (for Red, Blue, Green). You now have 2 × 3 = 6 paths through the diagram. STEP 3 — List the combined outcomes. Each path from start to end gives one combined outcome. Reading left to right: H-R, H-B, H-G, T-R, T-B, T-G. That is 6 outcomes in total. STEP 4 — Assign probabilities. Write the probability of each branch next to it. The coin is fair, so P(H) = 1/2 and P(T) = 1/2. The spinner has three equal sections, so P(R) = P(B) = P(G) = 1/3. To find the probability of a full outcome, multiply the probabilities along that path. This works because the two events are independent — the result of the coin does not affect the spinner. P(H then R) = P(H) × P(R) = 1/2 × 1/3 = 1/6. All six outcomes each have probability 1/6, and 6 × 1/6 = 1. The probabilities of all outcomes must always add up to 1 — this is a useful check.