Lesson Guide
Explanation
A multistage experiment happens in stages, like tossing one coin and then another. A tree diagram shows it. The first set of branches is the first coin, heads or tails, each with probability one half. From the end of each branch, a second set shows the second coin. Reading along the paths gives every outcome: heads heads, heads tails, tails heads and tails tails. Each path has probability one half times one half, which is one quarter. One head and one tail happens in two ways, heads tails and tails heads, so its probability is two quarters, or one half.
Your Turn
With the same two coins, find P(two tails). Check: only the path TT, so 1/4.
Watch Out
HT and TH are different paths. Listing just "two heads, one head, no heads" as three equal outcomes gives 1/3 for one head, which is wrong.
Explanation
Branches do not have to be equally likely. Suppose the chance of rain is zero point three on each of two days, so the chance of a dry day is zero point seven. The two branches from any point always add to one. To find the probability of one whole path, multiply along it. Rain then rain is zero point three times zero point three, which is zero point zero nine. Rain then dry is zero point three times zero point seven, which is zero point two one. Do all four paths, and check that they add to one.
Your Turn
If P(rain) were 0.1 each day, find P(rain on both days). Check: 0.1 × 0.1 = 0.01.
Watch Out
Multiply along a path, never add. 0.3 + 0.3 = 0.6 is not P(rain on both days); it is bigger than P(rain on one day), which cannot be right.
Explanation
Some events can happen along more than one path. Rain on exactly one day happens as rain then dry, or as dry then rain. Each of those paths has probability zero point two one. Because the event can happen either way, add the paths: zero point two one plus zero point two one is zero point four two. So the rule for every tree is: multiply along each path, then add the paths that fit the event.
Your Turn
Find P(rain on at least one day). Check: 0.09 + 0.21 + 0.21 = 0.51.
Watch Out
Counting only rain-then-dry gives 0.21. Exactly one rain day has two paths, so it is 0.42.
Explanation
A third stage adds a third set of branches. With three coins, each of the two branches splits into two, and each of those splits again, so there are two times two times two, which is eight paths. Each path has probability one half times one half times one half, which is one eighth. Exactly two heads happens on three paths: heads heads tails, heads tails heads, and tails heads heads. So the probability is three eighths.
Your Turn
Find P(all three coins land the same way). Check: HHH or TTT, so 2/8 = 1/4.
Watch Out
Two heads can come in any position. Counting only HHT gives 1/8; there are three such paths, so 3/8.
Explanation
At least one six means one six or two sixes. You could add the three paths that contain a six, but there is a quicker way. The only path with no six at all is not six, then not six, with probability five sixths times five sixths, which is twenty five over thirty six. Every other path has at least one six, so the probability is one minus twenty five over thirty six, which is eleven over thirty six. Quiz and exercises now.
Your Turn
A die is rolled three times. Find P(at least one 6). Check: 1 − 125/216 = 91/216.
Watch Out
1/6 + 1/6 = 1/3 is wrong: adding counts the path "6, 6" twice. Use 1 − P(no 6) = 11/36.
📌 Key Points — this lesson in a nutshell
- In a multistage experiment, a tree diagram shows every outcome: one set of branches per stage, with the probability written on each branch.
- Multiply along a path to get the probability of that one outcome.
- Add across all the paths that fit the event. Order matters on a tree: HT and TH are two different paths.
- The branches leaving any point add to 1, and the probabilities of all the end outcomes add to 1.
- At least one = 1 − P(none). It is usually much quicker than adding every path.
Step 1: Introduction
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