# What is the probability of having 3 double-yolk eggs in a dozen?

A friend of mine was making an omelet and found 3 double-yolk eggs in the box. What is the probability of that happening? (assume only 1 out of every 100 eggs is double-yolk)

Answer: 0.000205616 (probability of seeing 3 or more double-yolk eggs in a box)

Let’s make this question more interesting by changing it to: “what is the probability of having at least 2 double-yolk eggs in a box of dozen?”. For this, it would be in fact easier to find the probability of having only 1 or no double-yolk eggs in the box and then calculate one minus that probability.

$1 - Prob(only \nobreakspace 1 \nobreakspace regular \nobreakspace egg) - Prob(no \nobreakspace regular \nobreakspace egg)$

In the general form, if we want to have at least a minimum number of double-yolk eggs we can arrive at the following formula using the Binomial distribution:

$1 - \sum_{i=0}^{min \nobreakspace double-yolks} [P^i \times (1-P)^{(Box \nobreakspace Size - i)} \times {Box Size \choose i} ]$

If you solve this you will arrive at 0.000205616 as the answer.

To be sure that this formula is correct I wrote a little c++ code that could simulate this process. The results from the theoretical formula matches the simulation results.