Objects that look identical, such as two marbles of the same color, still need to be treated as individually distinct when building a probability sample space. This lesson explains why, and gives a reliable method — label everything, then group — for avoiding probability errors caused by unequal group sizes.
What you will learn
- Decide whether objects in a probability problem are distinguishable
- Adjust counts when identical objects are treated as indistinguishable
- Avoid over- or under-counting outcomes based on distinguishability
Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.
Distinguishable versus identical objects
Objects are distinguishable when each one can be told apart from the others, such as numbered tiles, named people, or differently colored balls. Objects are identical when nothing distinguishes them from each other, such as two marbles of the exact same color and size. Whether objects are described as identical usually refers to how they look, not to whether they are physically separate objects — two same-colored marbles are still two different marbles, even though they cannot be told apart by sight.
This distinction matters because a sample space must be built from outcomes that are genuinely equally likely. If two physically separate objects are merged into a single outcome just because they look alike, the resulting outcomes may no longer all be equally likely, which breaks the probability formula from the previous lesson.
Why the choice changes the count
Consider a bag with 2 red balls and 2 blue balls. If the two red balls are (incorrectly) treated as a single option and the two blue balls as another single option, it looks like there might be only a few color patterns to consider, and it becomes tempting to treat those patterns as equally likely without checking. In reality, there are 4 separate physical balls, and the number of ways to choose a pair of them depends on exactly which balls are available, not just their colors.
The danger is that patterns described only by color — such as "two red," "two blue," or "one of each" — usually do not occur equally often, even though there are only 3 such patterns to name. Counting from the individually labeled balls reveals the true, unequal sizes of these groups.
A reliable method: label everything, then group
The safest method is to always label every object individually first, even objects that look identical, such as calling two red balls and . Counting outcomes from this fully labeled sample space is always trustworthy, because every labeled object is genuinely distinct and every outcome built from them is equally likely.
Once the labeled sample space is built and counted, the objects can be regrouped by whatever pattern the problem actually asks about, such as color. Because the count came from the labeled sample space rather than an assumption, the resulting probability is guaranteed to be correct.
See the reasoning unfold
Pause after each step and predict what comes next.
Worked example 1
Two red balls from a mixed bag
A bag contains 2 red balls and 2 blue balls, all the same size and otherwise identical except for color. Two balls are drawn at random without replacement. What is the probability that both balls drawn are red?
- Even though the two red balls look identical, treat them as individually labeled to build a trustworthy, equally likely sample space.
1 of 3 steps revealed
Worked example 2
Why unlabeled color patterns aren't equally likely
Using the same bag of 2 red and 2 blue balls, explain why treating the three possible color patterns for a 2-ball draw — two red, two blue, and one of each — as three equally likely outcomes is incorrect, and find the correct probability of drawing one ball of each color.
- With balls labeled , the equally likely pairs are .
1 of 3 steps revealed
Worked example 3
Placing identical tokens into seats
Two identical tokens are placed into 2 of 3 numbered seats, chosen at random so that every pair of seats is equally likely to receive a token. What is the probability that seat 1 receives a token?
- Since the tokens are identical, an outcome here is simply the unordered pair of seats that receive a token; there are equally likely pairs: .
1 of 3 steps revealed
Common mistakes to catch
- Assuming that grouping outcomes by a visible pattern, like a color count, automatically gives equally likely groups, without checking how many labeled outcomes fall into each group.
- Failing to label physically identical objects individually when building a sample space, which can hide unequal group sizes.
- Using a combination count where the underlying labeled outcomes are not actually all equally likely to occur.
Try with support
Guided practice
Choose a strategy, take a first step, and use the hints when you need direction.
Guided problem 1
Difficulty 2/10A bag has 3 identical green marbles and 1 yellow marble. One marble is drawn at random. What is the probability that the marble drawn is yellow?
Guided problem 2
Difficulty 3/10A drawer has 2 identical black socks and 2 identical white socks. Two socks are drawn at random without replacement. What is the probability that both socks drawn are the same color?
Put it into practice
Independent practice
Work on your own first. If you reveal a solution, work through it and mark it reviewed.
Independent problem 1
Difficulty 3/10A box contains 3 identical red pens and 2 identical blue pens. Two pens are drawn at random without replacement. What is the probability that one pen of each color is drawn?
Independent problem 2
Difficulty 4/10Three identical coins are tossed. Assuming each individual coin lands heads or tails independently and each coin is fair, what is the probability that exactly 2 of the 3 coins land heads?
Independent problem 3
Difficulty 3/10A bag contains 4 identical blue chips and 1 red chip. Two chips are drawn at random without replacement. Which fraction gives the probability that both chips drawn are blue?
Independent problem 4
Difficulty 4/10A shelf has 2 identical copies of one novel and 3 identical copies of a different novel. Two books are chosen at random from the shelf without replacement. What is the probability that the two books chosen are different novels?
Bring it together
- Objects that look identical must still be labeled individually when building a sample space, so every outcome is equally likely.
- Grouping outcomes by a visible pattern is only valid once the labeled outcomes within each group have been counted.
- The same underlying labeled sample space can be reused to find the probability of many different color- or category-based events.
Your answers, revealed steps, and lesson progress are saved in this browser.