Distribution of female eggs

(Uses discrete probability)

Many models of bird populations are ‘female-only’ models; we keep track only of the number of females in the population, and assume the number of males is roughly the same. For large populations, we can use one-half of the average number of eggs per female as the average number of female eggs per adult female. For small populations, though, the actual distribution of numbers matters. Clearly, endangered species are more likely to be small populations.

The actual data we will have is a count of numbers of eggs in nests. Suppose we have

Number of eggs	0	1	2	3	4	5	6
Percentage of nests	3	2	5	12	44	29	5

Assuming that the probability of a given egg being female is 0.5, what is the probability, P(k), of having k female eggs in a given nest? The chart below gives probabilities of a number of female eggs, given the number of eggs. Students should be able to recognize a pattern, seeing Pascal’s triangle. More advanced classes may be able to derive this, or coin flipping can be used to approximate this.

	Number of female eggs
Eggs in nest	0	1	2	3	4	5	6
0	1	0	0	0	0	0	0
1	1/2	1/2	0	0	0	0	0
2	1/4	2/4	1/4	0	0	0	0
3	1/8	3/8	3/8	1/8	0	0	0
4	1/16	4/16	6/16	4/16	1/16	0	0
5	1/32	5/32	10/32	10/32	5/32	1/32	0
6	1/64	6/64	15/64	20/64	15/64	6/64	1/64

Combining these, the students should be able to come up with

Number of female eggs	0	1	2	3	4	5	6
Probability	0.015	0.24	0.325	0.231	0.084	0.014	0.001