Urn with replacement: A bag has 5 white, 7 red, and 8 black balls (20 total). Four balls are drawn one by one with replacement. What is the probability all four are white?

Introduction / Context:
With replacement, each draw is independent and identically distributed (i.i.d.). We want the probability all four outcomes are white from a fixed proportion.

Given Data / Assumptions:
White balls = 5, total = 20 ⇒ P(white on any draw) = 1/4; 4 i.i.d. draws.

Concept / Approach:
P(all white) = (P(white))^4 because of independence with replacement.

Step-by-Step Solution:

Verification / Alternative check:
Simulated reasoning: each draw 25% chance, the joint event multiplies four times.

Why Other Options Are Wrong:
1/16 corresponds to only two draws; fractions like 4/20 or 4/8 are single-draw proportions, not fourfold joint probabilities.

Common Pitfalls:
Forgetting that “with replacement” makes draws independent, leading to exponentiation of a single probability.

Final Answer:
1/256