### Problem S-12

## "P & Q" GRAPHING PROBLEM

### (To do BEFORE class on Friday, Nov. 11, and bring to class.)

This is a simple algebra and graphing
problem to begin our study of population genetics.

Back in Part I of the course, we used "p" and "q"
as the probabilities of two mutually exclusive events that add up to 1 (100%).
We used p=q=1/2 in the context of gender (girl/boy) calculations, and we used
p=3/4, q=1/4 for calculating dominant/recessive phenotype probabilities in specific-size
families from two parents who are both heterozygous for a simple Mendelian gene
*alpha*.

Now we are going to look at LARGE POPULATIONS rather than individuals,
and we are going to find relevance to the situations of p and q
being any values between zero and one, as long as p+q=1.

Starting from p+q=1,

squaring both sides gives ...... (p+q) squared =
1,

which is
*p squared + 2 p q + q squared = 1*

For all of the possible values of
p and q, in tenths, calculate each individual term in the binomial equation.
That is, start with {p=1, q=0}, and you get p squared = 1, 2pq=0, and q squared
=0. Now do the same for {p=0.9, q=0.1}, then for {p=0.8, q=0.2}, etc., down
to {p=0, q=1}.

For graphing, make the
horizontal axis be p (starting at zero and going up to 1 in tenths) {these same
horizontal axis points should also be labelled as q, starting at 1 and going DOWN to zero in tenths}.
Make the vertical axis be the calculated numbers for p squared, 2pq, and q squared
separately. Connect the points to give smooth curves for each of the three terms. By the nature of the basic algebra, the graph should have left-right symmetry
and, at any point along the horizonal axis, the three vertical values should
add up to one.

Solve algebraically for the exact values of p and q at points where the 2pq curve crosses another curve.

Bring your graph to class with you on Friday, November 11.