# MathBench>Population Dynamics

## Exact probabilities

Scribbling in the margin of his book, the Assistant soon had the answer his supervisor was waiting for. His reasoning went something like this...

For a given, single locus:

• let p be the proportion of dominant alleles (brown hair) in the gene pool
• let q be the proportion of recessive alleles (blue hair) in the gene pool

Using p and q, we can formalize the calculations you made a few pages ago. Remember the laws of AND and OR? Here are the same formulas, but when you roll the mouse over the graphics, you'll see the notation with p and q instead of numbers.

First, what is the probability that an offspring gets 2 blue alleles?

And the probability that the offspring will be all-brown (no blue allele)?

And finally, the probability that the offspring will be a blue carrier?

So to summarize, assuming the random mating occurs

• the proportion of homozygous dominant genotypes = p*p = p2
• the proportion of heterogeneous genotype (carriers) = p*q + q*p = 2pq
• the proportion of homozygous recessive genotypes = q*q = q2.
 "Neat-o" muttered the Assistant with self-aware irony. "Using p and q, I can avoid all the gobbledy-gook about probability. I just need to remember p2 for the dominant genotype, 2pq for the carriers, and q2 for the recessive type. Pretty simple. I bet if I tried, I could even forget where it came from..." But first, the Assistant shot off a triumphant email to the ME:

## Fill in the blanks (you may need a calculator)

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Sir: if 22% of the allele pool is the recessive gene
( in other words, p = and q = ), then, with 2 decimal places,

• % of the population would be expected to be carriers, and
• % would be expected to have blue hair.

In other words, rounded to no decimal places, I predict the following distribution:
% / % / % .

p.s., Did you know Google can be used as a calculator? Just type your equation with a equals sign,
i.e., .22 * .22 =, in the search box. Neat-o.

answers:0.78, 0.22, 34.32, 4.84, 61, 34, 5