A tree diagram is a branching diagram that shows all possible outcomes of a multi-step random process. Each branch represents one choice or event, and complete paths show combined outcomes.
A tree diagram prevents you from losing cases when a probability problem unfolds in stages. Instead of guessing the outcomes, you build them step by step.
Showing a random 20 of 50 problems.
Example 1
hard
A bag has 5 red and 5 blue balls. Two are drawn without replacement. Find P(at least one red).
Example 2
easy
A bag has 1 red and 3 blue balls. A tree shows one draw. What are the branch probabilities?One draw: P(Red)
Example 3
medium
A bag has 2 red, 2 blue, 1 green. One ball is drawn, then another with replacement. Find P(both red).
Example 4
medium
A test has two questions, each true/false. Using a tree, find P(both correct) by random guessing.P(both correct by guessing)
Example 5
hard
From the previous question, given a positive test, what is the probability of disease?
Example 6
challenge
A box has 2 red and 3 green. Two drawn without replacement. Use a tree to find P(one of each color).P(one of each) = P(RG) + P(GR)
Example 7
easy
In a tree diagram, what operation combines branch probabilities along one full path?
Example 8
easy
A spinner has 3 equally likely outcomes. A tree models spinning twice. How many complete paths?
Example 9
hard
Box 1 has 1 red and 3 blue balls. Box 2 has 2 red and 2 blue. A box is chosen at random (each 21), then one ball is drawn. Find P(red).P(red) = 1/2×1/4 + 1/2×2/4 = 1/8 + 2/8 = 3/8
Example 10
hard
On a two-stage tree, the path 'A then B' has probability 0.30. The branch probability for A is 0.6. What is the branch probability for B given A?
Example 11
medium
A spinner lands red 41 or blue 43, spun twice. Find P(both same color).P(both same) = P(RR) + P(BB)
Example 12
challenge
A bag has 3 red and 2 blue. Three drawn without replacement. Find P(all three red).
Example 13
medium
A bag has 1 red and 3 blue. Two drawn without replacement. Using a tree, find P(blue then blue).P(blue then blue)
Example 14
medium
A weather forecast says rain tomorrow has probability 0.4. If it rains, traffic jam probability is 0.7; if not, it's 0.2. Find P(traffic jam).P(traffic jam) = 0.4×0.7 + 0.6×0.2 = 0.28 + 0.12 = 0.40
Example 15
challenge
A factory has machines A and B making widgets. A makes 60% of widgets with 5% defective; B makes 40% with 2% defective. A widget is found defective. What is the probability it came from A?P(A | defective) = 0.030 / (0.030+0.008) ≈ 0.789
Example 16
easy
A path has branches with probabilities 21, 31, 41. What is the path probability?
Example 17
medium
Three children: each can be boy or girl with equal probability. How many complete paths and how many have at least two girls?
Example 18
hard
A disease has prevalence 0.01. A test gives a positive result with probability 0.99 if diseased, 0.05 if not. Find P(positive test).
Example 19
medium
Flipping two coins, find P(at least one head).P(at least one head) = P(HH)+P(HT)+P(TH)
Example 20
challenge
A spinner gives win 31 or lose 32, spun three times. Find P(exactly one win).