Introduction
Gregor Mendel's pea experiments, published in 1866, showed that heredity follows the rules of probability (Mendel 1925). A short table, now called a Punnett square, turns his idea into a prediction you can check against his counts.
This walkthrough builds the table for one cross, predicts counts for small and large batches of offspring, and compares the prediction with numbers Mendel reported.
The Starting Point
You have two pea plants. Each carries one version of the gene for yellow seeds and one version for green seeds. A gene is a unit of heredity, passed from parent to offspring, that influences a trait. Mendel didn't use that word, which came later, but the idea matches his model.
Yellow is dominant: a plant with one yellow version and one green version has yellow seeds. Green is recessive: it shows only when a plant carries two green versions. So both parent plants look yellow, even though each carries a hidden green version.
Write Y for the yellow version and y for the green. Each parent is Yy. The question is what colors their offspring will have, and in what proportions. In peas, each seed is a new offspring, so you can count offspring by counting seeds.
Walking Through the Cross
Step 1: List what each parent can pass on
Each offspring receives one version of the gene from each parent. A Yy parent passes on either Y or y, and the two are equally likely. So each parent passes Y with probability 1/2 and y with probability 1/2.
The two parents act independently. What one parent passes on doesn't change what the other passes on. That independence is what lets you multiply probabilities in the next step.
Step 2: Build the four-cell table
Put one parent's possible versions across the top and the other parent's down the side. Each cell combines one version from each parent.
| Y (from parent 1) | y (from parent 1) |
|---|
| Y (from parent 2) | YY | Yy |
| y (from parent 2) | Yy | yy |
Each cell has a probability of \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. The four cells are equally likely, and together they cover every possibility, so their probabilities add up to 1.
Step 3: Count the colors
Read the table cell by cell:
- YY: one cell, pure yellow
- Yy: two cells, mixed, which look yellow because yellow is dominant
- yy: one cell, pure green
The genetic types come in the pattern 1 to 2 to 1. Mendel's own formula for the offspring of hybrids, written A + 2Aa + a, records the same pattern (Mendel 1925). The colors, though, come in the pattern 3 to 1, because three of the four cells contain at least one Y. So the chance that an offspring has yellow seeds is 3/4, and the chance that it has green seeds is 1/4.
Step 4: Predict counts for 100 and for 8,000 offspring
Expected counts come from multiplying the number of offspring by each probability.
| Offspring | Expected yellow | Expected green |
|---|
| 100 | 100 \times \frac{3}{4} = 75 | 100 \times \frac{1}{4} = 25 |
| 8,000 | 8{,}000 \times \frac{3}{4} = 6{,}000 | 8{,}000 \times \frac{1}{4} = 2{,}000 |
Which batch is likelier to land near 3 to 1? The law of large numbers says that as the number of trials grows, the fraction of times an outcome happens settles toward its probability. So the larger batch should come closer.
The numbers bear this out. Take "near 3 to 1" to mean that between 73 and 77 percent of offspring are yellow. A batch of 100 lands in that window only about 44 percent of the time. A batch of 8,000 lands in it almost every time, with a chance above 99.99 percent. The count of yellow offspring varies more in absolute terms in the larger batch, but as a fraction of the batch it varies far less.
Step 5: Compare with Mendel's counts
In his experiment on seed color, Mendel let hybrids of exactly this kind fertilize themselves, which gives the same four-cell table. He reported that 258 plants yielded 8,023 seeds, of which 6,022 were yellow and 2,001 green, a ratio he gave as 3.01 to 1 (Mendel 1925, Expt. 2).
The prediction for 8,023 seeds is 8{,}023 \times \frac{3}{4} \approx 6{,}017 yellow and about 2,006 green. Mendel's count is off by about 5 seeds in each color. His yellow share was about 75.06 percent, against a predicted 75 percent.
Single plants told a different story. Mendel reported one plant with 32 yellow seeds and 1 green, and another with 20 yellow and 19 green (Mendel 1925, Expt. 2). Each is a small sample, and each is far from 3 to 1. The ratio emerges in the total, not in every pod.
Key Considerations
Terms then and now. Mendel wrote of "differentiating characters," not genes, and the English translation keeps his terms "dominant" and "recessive." He described the seed color as the color of the seed albumen. The table works the same in either vocabulary.
The conditions the prediction needs. The 3-to-1 prediction assumes that each parent passes each version with probability 1/2, that the parents act independently, and that the trait depends on one gene with a dominant and a recessive form. Mendel chose traits that come in two distinct forms. Many traits don't work this way, and they don't follow simple ratios.
The common mistake: expecting exactly 3 to 1 in a small sample. A ratio of 3 to 1 means each offspring has a 3/4 chance of yellow. It doesn't mean that every group of four offspring has exactly three yellow and one green. With four offspring, all four yellow is quite possible, and so is two and two. Mendel's plant with 20 yellow and 19 green seeds isn't evidence against the model. It's what you should expect to see sometimes in a sample that small. The ratio is a statement about the long run, and only large totals should be expected to land close to it.
Summary
Crossing two carriers gives four equally likely combinations, three of which show the dominant color, so the model predicts 3 yellow to 1 green. Mendel's large total matched that prediction closely, and his single plants did not.
| Y | y |
|---|
| Y | YY (yellow) | Yy (yellow) |
| y | Yy (yellow) | yy (green) |
| Sample | Predicted yellow : green | Observed yellow : green |
|---|
| 8,023 seeds (258 plants) | about 6,017 : 2,006 | 6,022 : 2,001 (3.01 to 1) |
| One plant | about 3 to 1 | 32 : 1 |
| Another plant | about 3 to 1 | 20 : 19 |
- Each cell of the table has probability 1/4, and three cells give yellow, so the chance of yellow is 3/4.
- The large total lands within about 5 seeds of the prediction, as the law of large numbers leads you to expect.
- Single plants vary widely, so a small sample far from 3 to 1 doesn't contradict the model.
References
- Mendel, Gregor. 1925. Experiments in Plant-Hybridisation. Translated for the Royal Horticultural Society, London (1901); with William Bateson. Cambridge, MA: Harvard University Press.