Why the ratio is 1 : 2 : 1
Why is the middle number of the ratio a two? Because there are two different ways to end up with one of each factor, and only one way to end up with a matching pair.
The four equally likely outcomes
Each parent contributes one gamete, and each gamete is equally likely to carry either factor. That gives four combinations, each with the same chance of happening.
| From the ovum | From the pollen | Offspring | Chance |
|---|---|---|---|
| T | T | TT | 1 in 4 |
| T | t | Tt | 1 in 4 |
| t | T | Tt | 1 in 4 |
| t | t | tt | 1 in 4 |
Two of the four rows give the same pair
Rows two and three both produce Tt. They are genuinely different events — in one the dominant factor came through the ovum, in the other through the pollen — but the offspring is the same either way. Two chances in four is a half, which is why Tt is twice as common as either of the others.
And that is how 1 : 2 : 1 becomes 3 : 1
Three of the four rows contain at least one T, so three in four plants are tall. The appearance ratio is not a separate fact; it is the factor ratio with the first two groups added together, because you cannot tell them apart by looking.
The ratio is about very large numbers
One in four does not mean that in any four seeds exactly one will be dwarf. Each seed is a fresh throw. A family of four children can easily be four boys, and a pod of four peas can easily be four tall.
Two ways in, twice as likely
Tt can be made in two ways and TT or tt in only one each. That single fact is what makes the genotype ratio 1 : 2 : 1 and the appearance ratio 3 : 1.