Returning the first token changes what is available next. Consider a hypothetical container with red R1, red R2 and blue B1. Each draw is uniform among available tokens. Compare leaving the first token out with returning it and independently reshuffling before drawing again.
What replacement changes
Hypothetical replacement model: after the first draw, the token is returned and the container is reset and reshuffled independently. The second draw again has R1, R2, and B1 available. Therefore, after a red first draw, the probability of red on the second draw is 2/3.
Hypothetical no-replacement model: the first token stays out. If the first draw was red, one red token has gone, leaving one red token and one blue token. The second-draw probability of red is then 1/2.
Independence leaves the next-event probability unchanged after conditioning; see OpenStax’s explanation of independent events.
Counting both-red outcomes
| Model | Favourable ordered pairs | Total ordered pairs | Probability |
|---|---|---|---|
| With replacement | 4 | 9 | 4/9 |
| Without replacement | 2 | 6 | 1/3 |
With replacement, the first position can be R1 or R2, and the second position can independently be R1 or R2. The four favourable pairs are R1,R1; R1,R2; R2,R1; and R2,R2, out of 3 × 3 = 9 ordered pairs. Equivalently, the calculation is (2/3) × (2/3) = 4/9.
Without replacement, the favourable pairs are only R1,R2 and R2,R1. R1,R1 and R2,R2 are impossible because a drawn token is not returned. There are 3 × 2 = 6 ordered pairs, giving 2/6 = 1/3, also equal to (2/3) × (1/2).
The practical reading rule
First state whether the pool resets. Then define the available tokens before each draw, preserve their labels when counting, and use ordered pairs when first and second positions matter. Replacement permits the same labelled token to appear twice; no replacement does not. The numerical result comes from that explicit hypothetical assumption, not from a screen label or from evidence about any real game mechanism.
For a different question about interpreting past observations, see our history-screen reading note. The hypothetical counts above are not observed results.