Sampling colored counters without replacement
A classroom bag contains five red counters and three blue counters, all identical except for color. Two counters will be drawn randomly without replacement. Before the experiment, students must calculate the probability that both counters are red and compare it with the result for an otherwise identical experiment with replacement. The bag is thoroughly mixed before the first draw, and every counter is equally likely to be selected at each permitted draw. The teacher records ordered draws only to make the conditional reasoning visible; no extra counter is added. A student argues that the second probability must still be five eighths because the bag originally contained five red counters. Another treats the two colors as equally likely merely because there are two colors.
1. Find the no-replacement probability of two reds.
Read the worked answer
The first red probability is 5/8. After a red is removed, the next red probability is 4/7. Their product is 20/56=5/14.
2. Compare replacement.
Read the worked answer
With replacement and independent mixing, the probability is (5/8)²=25/64. The denominator and numerator change in the no-replacement case; two colors do not imply equal probabilities.