Probability Tree Diagram Without Replacement

Learn why each draw changes the next probability, then solve a tree one branch at a time.

Core idea

What Does Without Replacement Mean?

Without replacement means that the item you draw stays out of the collection. The next draw is made from what remains, so its probability depends on the first draw.

That is why a tree diagram without replacement uses conditional probabilities on later branches instead of repeating the first set of fractions.

Conditional probability

P(A then B) = P(A) × P(B | A)

P(B | A) is the conditional probability of B after A has already happened. Without replacement, that second probability usually changes because one item has been removed.

How to Draw a Tree Diagram Without 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.
2Update what remains on each branch
Treat each first-draw branch separately. Remove the item drawn on that branch, so the total decreases by one and that result’s count may decrease too.
3Add the second branches
From each updated collection, draw every possible second result. Use the remaining total and remaining counts when writing these probabilities.
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 red and one blue

Use the four steps above to build the tree and solve the question one action at a time.

A bag has 3 red and 2 blue marbles. Draw two marbles without replacement. What is the probability of one red and one blue?

Start: 3 red, 2 blue — 5 total.

1First draw
2Update counts
3Second draw
4Routes

Worked Examples

Worked example

Counters: 4 black and 3 white

Problem

A box contains 4 black counters and 3 white counters. Two counters are drawn one after another without replacement.

Find

Find the probability of drawing exactly one black counter and one white counter.

Worked example

Class representatives: 3 girls and 2 boys

Problem

A class has 3 girls and 2 boys. Choose a president and then a vice-president without replacement.

Find

Find the probability that the president is a girl and the vice-president is a boy.

Worked example

Standard deck: 4 aces and 48 non-aces

Problem

A standard 52-card deck contains 4 aces and 48 non-aces. Draw two cards one after another without replacement.

Find

Find the probability of drawing at least one Ace.

Common Mistakes

Keeping the original denominator

Without replacement, one item is gone after the first draw. The second denominator must be one less than the first.

Not updating the drawn category

On each branch, reduce the count only for the category drawn on that branch before writing the next probability.

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 box has 5 black and 3 white counters. Draw two without replacement. Find P(two black).

Show answer

5/8 × 4/7 = 5/14

From 4 seniors and 2 juniors, choose a captain then a vice-captain. Find P(exactly one junior).

Show answer

4/6 × 2/5 + 2/6 × 4/5 = 8/15

A bag has 2 gold and 6 silver tokens. Draw two without replacement. Find P(at least one gold).

Show answer

1 − (6/8 × 5/7) = 13/28