Mathematics, four units

What changes in the calculation when you draw without replacement?

Quick answer

When you draw without replacement, the total count of items decreases after every pick, reducing the denominator by one each time. If the selected item matches the target category, that category's count also drops, reducing the numerator by one. Because earlier outcomes alter future possibilities, the draws are dependent rather than independent.

The topic explained

How the fraction changes on each draw

When you draw without replacement, an item taken from the pool is not returned before the next selection. This changes the conditions for every subsequent event. Because the total number of items is now smaller, the denominator of your probability fraction must decrease by one on each consecutive draw.

The numerator also changes if you draw from the specific group you are tracking. For example, if you hold an urn with 5 red marbles and 3 blue marbles (8 total), the probability of choosing a red marble first is 5/8. If you keep that red marble out, only 4 red marbles remain out of 7 total marbles. The second probability becomes 4/7. To find the probability of both events occurring, you multiply these updated fractions: (5/8) * (4/7) = 20/56, which simplifies to 5/14.

With replacement versus without replacement

With replacement, the item is returned to the pool, resetting the experiment to its original state. Every draw remains independent, and the probabilities stay identical. Drawing two reds with replacement from the same urn gives (5/8) * (5/8) = 25/64 (approximately 0.391), whereas drawing without replacement gives 20/56 (approximately 0.357). Removing items without returning them directly lowers the chance of repeating that same outcome.

Because the probability of success shifts from trial to trial when you do not replace items, the trials are not independent. This means you cannot use the standard binomial distribution formula for draws without replacement. The binomial model strictly requires that the probability of success remains constant across all trials.

Shortcuts and mistakes to avoid

A frequent mistake is reducing the numerator while forgetting to reduce the denominator, writing 4/8 instead of 4/7. Both totals shrink whenever an item leaves the pool.

When a problem asks for the probability of getting 'at least one' item of a certain type across multiple dependent draws, calculating every matching combination directly can be long and prone to errors. Instead, use the complement rule: subtract the probability of getting zero matching items from 1. Calculating 1 minus P(none) takes only one line of multiplication.

Free printable worksheet

Probability of Draws Without Replacement

For 11th-grade math students. Calculate the probability for each scenario where items are drawn without replacement. Give answers as simplified fractions.

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Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Probability of Draws Without ReplacementName:

A bag contains 4 red balls and 6 green balls. You draw one ball, keep it, and draw a second. What is the probability that both balls are red?
A drawer holds 8 black socks and 4 blue socks. You pull out two socks at random without replacement. What is the probability that both are black?
A box contains 5 green tokens, 3 yellow tokens, and 2 white tokens. Two tokens are drawn without replacement. What is the probability the first is green and the second is yellow?
From a standard deck of 52 cards, you draw two cards without replacement. What is the probability of drawing two Aces?
A bin has 7 working light bulbs and 3 defective ones. You choose two bulbs without replacement. What is the probability of selecting at least one defective bulb?
A committee of 3 students is chosen at random without replacement from a group of 6 girls and 4 boys. What is the probability that all 3 selected are girls?

Common questions

Why cannot I use the binomial formula when drawing without replacement?

The binomial formula requires independent trials with a constant probability of success. Drawing without replacement changes the contents of the pool after each draw, which causes the probability of success to change from trial to trial.

What happens if I draw an item of a different type on the first draw?

The denominator still decreases by one because the total pool is smaller, but the numerator for your target category remains unchanged since none of those specific items have been removed yet.

How do two-way tables connect to dependent probability?

A two-way table organizes counts across two categories. When calculating conditional probability from a table, you restrict your focus to a single row or column total as your denominator instead of the grand total.

Is drawing cards from a deck always dependent?

Yes, as long as the drawn cards are not placed back into the deck and reshuffled before the next draw.

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