Independent events
P(A then B) = P(A) × P(B)
With replacement, B is independent of A: the drawn item is returned, so P(B) stays the same no matter what happened on the first draw.
Probability Tree Diagram With Replacement
Learn why each probability stays the same on every draw, then solve a tree diagram with replacement one route at a time.
Core idea
What Does With Replacement Mean?
With replacement means that the item you draw goes back into the collection before the next draw. The collection is fully restored, so every draw faces exactly the same probabilities.
That is why a tree diagram with replacement repeats the first set of fractions on the second branches: the draws are independent events, and no counts ever change.
How to Draw a Tree Diagram With Replacement
1Draw the first branches
Start with the full collection. Draw one branch for every possible first result, and write each probability as number of that result ÷ total items.
2Return the item after each draw
On every first-draw branch the drawn item goes back into the collection. The total and every category count are exactly what they were at the start.
3Repeat the first-draw probabilities
Add the second-draw branches using the same fractions as the first draw. The branches under each parent still add up to 1.
4Use the routes to answer the question
Multiply probabilities along a complete route. When the same event can happen in several different orders, add the probabilities of all matching mutually exclusive routes.
Guided walkthrough
Guided example: One gold and one silver
Use the four steps above to build the tree and solve the question one action at a time.
A bag has 3 gold and 2 silver tokens. Draw two tokens with replacement. What is the probability of one gold and one silver?
Start: 3 gold, 2 silver — 5 total.
- 1First draw
- 2Return item
- 3Second draw
- 4Routes
Worked Examples
Worked example
Marbles: 6 red and 4 blue
Problem
A bag contains 6 red marbles and 4 blue marbles. Two marbles are drawn one after another with replacement: the first marble is returned before the second draw.
Find
Find the probability of drawing at least one blue marble.
Worked example
Spinner: 3 green and 5 non-green sectors
Problem
A fair spinner has 8 equal sectors: 3 green and 5 non-green. The spinner is spun twice. Like drawing with replacement, no sector is removed, so the second spin has the same probabilities as the first.
Find
Find the probability of landing on green exactly once.
Worked example
Word tiles: 3 letter-A and 3 other tiles
Problem
A set of 6 letter tiles spells BANANA: 3 tiles show A and 3 show B or N. Draw two tiles one after another with replacement: the first tile goes back before the second draw.
Find
Find the probability of drawing at least one A tile.
Common Mistakes
Lowering the second denominator
With replacement the item goes back, so the total stays the same: the second denominator equals the first.
Treating the draws as dependent
Because the full collection is restored, the first result cannot change the second draw. The two draws are independent events.
Forgetting the other order
For one of each type, include both possible orders unless the question specifies an order, such as first then second.
Adding overlapping at-least-one paths
For at least one target item, the complement is often simpler: find the chance of no target items, then subtract from 1.
With vs. Without Replacement
| Topic | With replacement | Without replacement |
|---|---|---|
| Returned item | Returned to the collection | Removed from the collection |
| Second-draw denominator | Same as the first draw | One less than the first draw |
| Event relationship | Independent | Dependent |
| Calculation | Probabilities stay the same | Use probabilities conditional on the first draw |
Practice Questions
A drawer has 5 black and 3 white pens. Take two with replacement. Find P(two black).
Show answer
5/8 × 5/8 = 25/64
A card is drawn from a standard deck, returned, and drawn again. Find P(both hearts).
Show answer
13/52 × 13/52 = 1/16
A jar holds 2 gold and 6 silver beads. Pick two with replacement. Find P(at least one gold).
Show answer
1 − (6/8 × 6/8) = 7/16
Probability Tree Diagram With Replacement FAQ
Quick answers about independent draws and probability trees.
- How to determine probability with replacement?
How to determine probability with replacement?
Divide the number of wanted objects by the total, return the drawn object, then multiply the matching path probabilities, which stay the same on every draw.
- Can you provide some examples of probability tree diagrams?
Can you provide some examples of probability tree diagrams?
This page uses marbles, a spinner, and letter tiles to show one path, paths in either order, and a complement.
- What is probability with and without replacement?
What is probability with and without replacement?
With replacement restores the collection so later events are independent. Without replacement changes the collection so later events are dependent.
- Why do the probabilities stay the same after the first draw?
Why do the probabilities stay the same after the first draw?
The first object is returned before the second draw, so the collection is unchanged and the second set of branches repeats the first fractions.
Build Your Own Probability Tree
Use the calculator to model your next probability question.