Imagine that we flip a coin, where if it comes up heads, we lose three ships, and if it's tails, we lose no ships. That would get us the 1.5 average we're looking for, but it implies there's a 50% chance of losing three ships, a 50% chance of losing no ships, and a 0% chance of any other number. Not very realistic.

Instead, let's flip three coins and count each head as a sunken ship. Then we have a 1/8 chance of losing zero or three, and a 3/8 chance of losing one or two. That's better...but it implies that the chance of losing four or more is exactly zero. The probability of losing four or more ships is probably lower than the probability of losing any fewer number, but it's likely to be more than zero.

So, we could flip six coins and count each head as half a sunken ship. Or, flip twelve coins and count each head as a quarter of a sunken ship. If we continue to add flips and weigh them less, then we start to have the possibility of sinking more ships than we have. But, the numbers from zero to, say, six or seven look very realistic.

That's the Poisson distribution, and it's useful any time we want to estimate the probability a rare event occurs k times, if on average it would occur λ times.

The formula is .