Mathwords logoMathwords

Sampling with Replacement — Definition, Formula & Examples

Sampling with replacement is a way of selecting a sample where you put each chosen item back into the population before the next draw. The same item can be picked more than once.

A sampling procedure in which each selected unit is returned to the population before the next selection. For a population of size NN, every draw has the same NN possible outcomes, so the draws are independent and identically distributed. Any unit may appear in the sample multiple times.

Key Formula

Number of ordered samples=Nn\text{Number of ordered samples} = N^n
Where:
  • NN = number of items in the population
  • nn = number of draws (the sample size)

Worked Example

Problem: A bag holds 5 marbles: 2 red and 3 blue. You draw 2 marbles with replacement. What is the probability that both are red?
Step 1: Count all possible ordered outcomes. Each draw has 5 options because the first marble goes back into the bag.
52=255^2 = 25
Step 2: Count the outcomes where both marbles are red. There are 2 red choices on each draw. Divide by the total number of outcomes.
P(both red)=2⋅225=425=0.16P(\text{both red}) = \frac{2 \cdot 2}{25} = \frac{4}{25} = 0.16
Step 3: For contrast, compute the same probability without replacement. Only 1 red marble remains out of 4 after the first red draw, so the second draw depends on the first.
25⋅14=110=0.10\frac{2}{5} \cdot \frac{1}{4} = \frac{1}{10} = 0.10
Answer: With replacement, P(both red)=425=0.16P(\text{both red}) = \frac{4}{25} = 0.16. Without replacement, the probability drops to 0.100.10.

Why It Matters

Binomial probability in AP Statistics requires independent trials. Replacing each item guarantees that independence. Real surveys sample without replacement, so AP Stats uses the 10% condition: if the sample is at most 10% of the population, you can treat the draws as approximately independent. Bootstrap methods in college statistics also rely on resampling your data with replacement.

Common Mistakes

Mistake: Multiplying the same probability on every draw when items are not replaced.
Correction: Without replacement, the population shrinks after each draw, so the probabilities change. Use the constant-probability approach only when items are replaced, or when the 10% condition is met.

Related Terms