Two-step Experiments: Sample Spaces, Trees and Tables

A two-step experiment has two things happening, like tossing two coins. The sample space is HH, HT, TH and TT — four outcomes, because there are two possibilities for the first coin and two for the second. So the probability of two heads is one quarter, and the probability of one of each is 2 over 4, which is one half. The critical point is that HT and TH are different outcomes: first coin head and second tail is not the same event as first tail and second head. The classic error is listing only three outcomes — two heads, two tails, one of each — which makes two heads look like one third. It is one quarter. Keeping the steps in order keeps the outcomes separate.

When both steps have several outcomes, a two-way table is the clearest tool. Put one die across the top, the other down the side, and write the result in each cell. Two dice give 6 times 6, which is 36 outcomes. Counting the cells, six of them total 7, so the probability of a sum of 7 is 6 over 36, or one sixth. A sum of 2 happens in just one cell, so 1 over 36. This shows something important: the 36 grid squares are equally likely, but the eleven possible sums are not — 7 comes up far more often than 2 or 12. So always count grid squares, never sums.

A tree diagram is the other main tool. Start at a dot on the left. The first set of branches shows step one, and from the end of each of those, a second set shows step two. Follow any path all the way to the right and you have one outcome. Two coins give four paths: HH, HT, TH and TT. The strength of a tree is that it keeps the order visible, which is exactly why HT and TH stay separate — they are different paths through the diagram. Trees work best when the two steps are different from each other, or when the second step depends on what happened in the first.

Two-step questions come in two flavours, and the wording tells you which. With replacement means the first item goes back, so the second draw has the same choices as the first — 5 marbles then 5 again, giving 25 outcomes. The two draws are independent. Without replacement means the first item stays out, so the second draw has one fewer — 5 then 4, giving 20 outcomes, and the draws are dependent. Look for the phrases: 'replaced', 'put back' or 'returned to the bag' mean with; 'not replaced', 'kept' or 'eaten' mean without. And 'takes two at once' counts as without, since you cannot pick the same marble twice. Underline the phrase before you draw anything.

Three tools, and the question tells you which fits. Use a simple list when there are only a few outcomes, say four to eight, like two coins. Use a two-way table when both steps have several outcomes, like two dice or two spinners. Use a tree diagram when the steps are different from each other, or when one depends on the other, as with marbles drawn without replacement. Whichever tool you pick, the final three steps are identical: count the total number of outcomes, count the favourable ones, then write the fraction and simplify. Always show the diagram, because it earns marks even if your final fraction slips.