Probability and tree diagrams
Maths · GCSE · The School
Along and across
A tree diagram has one rule for each direction. Multiply ALONG a set of branches to get the probability of that whole path. Add ACROSS the completed paths that satisfy what you were asked. Every pair of branches from one point must sum to 1, which is a free check on your working before you go any further.
Independent or not
Two events are independent when the first does not change the second: rolling a die twice, or tossing a coin twice. They are dependent when it does — taking two sweets from a bag WITHOUT replacing the first, because the bag now holds one fewer sweet and one fewer of that colour. Read the question for the words "without replacement"; they change every probability on the second set of branches.
At least one
Questions asking for the probability of AT LEAST ONE are usually far quicker done backwards. The opposite of at least one is NONE, which is a single path. Work out the probability of none and subtract from 1. Adding up every path with one, two or three of the thing is the slow route and the one where a path gets missed.
A bag holds 5 red and 3 blue counters, and you take two without replacing the first. FIRST DRAW: 5 out of 8 red, 3 out of 8 blue. SECOND DRAW, if the first was red: only 4 red left and 7 counters in total, so 4/7 red and 3/7 blue. If the first was blue: 5/7 red, 2/7 blue. Now the question: what is the probability of one of each? Two paths satisfy it. Red then blue is 5/8 × 3/7 = 15/56. Blue then red is 3/8 × 5/7 = 15/56. Add them: 30/56, which simplifies to 15/28. Notice the denominator changed from 8 to 7 on the second draw — that single change is what "without replacement" means, and forgetting it is the commonest lost mark on this topic.