Tree diagrams organize multi-step probability situations by showing each possible outcome as a path of branches. This cheat sheet helps students decide whether events are independent or dependent and then label branches correctly. It is useful for problems with coins, spinners, cards, marbles, surveys, and choices made in stages.
A clear tree diagram reduces missed outcomes and helps students calculate probabilities accurately.
The most important idea is that each complete path represents a sequence of events. To find the probability of one path, multiply the probabilities on its branches, such as . For dependent events, the second probability changes after the first event, so use .
To find the probability of several acceptable paths, add the probabilities of those paths.
Key Facts
- In a tree diagram, each branch shows one possible outcome and is labeled with its probability, such as .
- The probabilities on all branches leaving the same point must add to .
- For independent events, the outcome of the first event does not change the probability of the second event, so .
- For dependent events, the outcome of the first event changes the probability of the second event, so .
- The probability of a complete path is found by multiplying branch probabilities: .
- For independent events, the multiplication rule becomes .
- To find the probability of either of two non-overlapping paths, add their probabilities: .
- When drawing without replacement, update the total number of items and the number of favorable items after each draw.
Vocabulary
- Tree diagram
- A branching diagram that shows all possible outcomes of a multi-step probability experiment.
- Independent events
- Events are independent when the result of one event does not change the probability of another event.
- Dependent events
- Events are dependent when the result of one event changes the probability of another event.
- Conditional probability
- Conditional probability is the probability of an event happening given that another event has already happened, written as .
- Branch probability
- A branch probability is the probability written on one branch of a tree diagram.
- Path probability
- A path probability is the product of all branch probabilities along one complete path of a tree diagram.
Common Mistakes to Avoid
- Treating dependent events as independent is wrong because probabilities can change after the first outcome, especially when items are not replaced.
- Forgetting to update the denominator after drawing without replacement is wrong because the total number of items decreases after each draw.
- Adding branch probabilities along one path is wrong because a sequence of events uses multiplication, such as .
- Multiplying probabilities from different paths is wrong because separate acceptable outcomes are combined by addition, not multiplication.
- Leaving branch probabilities unlabeled is a mistake because the diagram cannot be checked or used reliably without every branch probability.
Practice Questions
- 1 A bag has red marbles and blue marbles. One marble is drawn, replaced, and then a second marble is drawn. Use a tree diagram to find .
- 2 A box has green cards and yellow cards. Two cards are drawn without replacement. Use a tree diagram to find .
- 3 A coin is flipped and a spinner with equal sections labeled , , , and is spun. Use a tree diagram to find .
- 4 A student draws two names from a hat without replacement. Explain why the second draw is dependent on the first draw and how the tree diagram should show that change.
Understanding Tree Diagrams for Dependent and Independent Events
Dependence is about what information is available before the next choice. If a card is removed from a deck, the deck itself has changed. If a coin is flipped, the coin has not changed because of the earlier flip.
This is the physical reason many school examples differ. Words in a question often give the clue. Without replacement, removed, chosen already, and remaining usually signal dependence.
With replacement, returned, reset, or repeated under the same conditions usually signal independence. Do not decide from the objects alone. Drawing marbles can be independent if each marble is returned before the next draw.
Consider a bag with three red counters and two blue counters. If one counter is taken and kept out, a first red result has probability three fifths. After that result, only four counters remain, with two red and two blue.
The chance of red followed by blue is three fifths multiplied by two fourths, which equals three tenths. A blue followed by red result is a different ordered outcome. Its probability is two fifths multiplied by three fourths, which is also three tenths.
If the question asks for exactly one red counter in two draws, both orders are acceptable. Their probabilities combine to give three fifths. This example shows why wording such as exactly one matters.
Independent events appear in repeated spins, rolls of a fair die, and computer random number generators that reset for each trial. Replacement creates the same effect in many classroom problems. In real situations, independence can be less obvious.
A survey of two people from the same household may produce related answers because their experiences are connected. Selecting two students for different teams may be dependent if the first student cannot be selected twice.
Mathematical independence does not mean two events have nothing in common in everyday language. It means that knowing one result gives no change to the numerical chance of the other result.
Careful checking catches many errors. At every split, the possible next outcomes should account for the whole situation. If they do not total one, a case may be missing or a denominator may be wrong.
Keep denominators tied to the number of items currently available, not the number at the start. Read whether order matters. Red then blue differs from blue then red as a sequence, though a question about one of each color includes both.
For phrases such as at least one success, it is often simpler to find the chance of no successes and subtract that result from one. Finally, list the requested end results before calculating. This prevents adding paths that sound similar but do not satisfy the condition.