Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 8 / a

Assuming the egg colours are independent with the same probability $p$, the number of white eggs has the binomial distribution with four trials. The possibilities are

The binomial coefficients count which eggs are white. Marginal colour probabilities alone would not determine nest probabilities without the independence assumption.

For $p=3/4$, the five probabilities in increasing white-egg order are $(1,12,54,108,81)/256$. The largest is $108/256=27/64$, so the most common nest has three white eggs and one brown egg, $W_3{B}_1$.