Simulating crosses with random numbers

A square is not a field

A Punnett square says one in four plants of Tt with Tt is short. It does not say that every four plants hold exactly one short plant. Each seed is a separate chance event, like a coin toss.

Mendel saw this. His counts were close to 3:1, but never exactly 3:1. A computer can show why, by making thousands of crosses in a second.

One child at random

random.choice picks one item from a string or list, each equally likely. A gamete is one letter chosen at random from a parent:

Each call gives a new child: sometimes TT, sometimes Tt, sometimes tt.

Many children

A generator inside Counter makes and counts many children at once:

The _ is a loop variable that is not used. With 100 plants you might see 23 short, or 29, or 21. Each run is different.

A seed for the same numbers

random.seed(3) at the top makes the "random" numbers start from the same place every run. Then you, a classmate and a teacher all see the same output, which makes checking easy. Remove the seed and each run differs.

Small numbers wander, large numbers settle

With random.seed(3), the program at the end of this lesson prints:

• 20 plants: $35.0$ % short;
• 100 plants: $23.0$ % short;
• 1000 plants: $26.2$ % short;
• 10000 plants: $25.4$ % short.

The share jumps about for a few plants and settles near 25 % for many. This is the law of large numbers. It is why Mendel counted hundreds of plants, not a handful.

How far does a count usually wander? For $n$ plants with a chance $p$ of being short, the typical wander is $\sqrt{n\,p\,(1 - p)}$ plants. It grows with $n$, but slower than $n$, so the share wanders less and less.

Observed against expected

A bar chart puts the two side by side. The expected counts come from the ratio: for $n$ plants, $\dfrac{3n}{4}$ tall and $\dfrac{n}{4}$ short.

Shifting the second set of bars by $0.8$ puts each pair side by side. When the bars are close, the data agree with the 3:1 rule.

Practise with new numbers

Try it in Python

The program crosses Tt with Tt for 20, 100, 1000 and 10000 plants with a fixed seed. It prints the share of short plants each time, then draws observed against expected for the last run.

Common mistakes

• Taking both letters from the same parent. random.choice(mum) + random.choice(mum) is a cross of mum with herself, not with dad.
• Calling the 3:1 rule wrong after a few plants. A small count can be far off. Simulate many more before you judge.
• Expecting a seed to make the numbers less random. It only makes them repeat. The pattern of chance is the same.
• Comparing counts from runs of different sizes. Compare shares, in percent, or scale the expected counts to each run.

Remember these

• random.choice(parent) picks one gamete; one choice from each parent makes a child.
• Counter(baby(...) for _ in range(n)) makes and counts n children.
• random.seed(k) repeats the same random numbers on every run.
• A few plants wander far from the ratio; many plants settle near it.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why can 8 plants from Tt with Tt include 4 short ones, when the rule says 1 in 4?

2. What does random.seed(3) change, and what does it not change?

3. A simulation of 10000 plants gives 25.4 % short. Is that a problem for the 3:1 rule?

4. Why does the bar chart draw the expected bars at 0.8 and 2.8 instead of 0 and 2?

5. Which line simulates one child of Tt with tt correctly?