What are the odds for a “19” (zero pt) hand in Cribbage?
I wrote a software simulation (20 million deals) to work this out, because the maths is far too complicated. The chances of getting a zero-point hand of five cards (assuming the starter card is chosen randomly among those five) is roughly 7.4%.
However, this assumes that your five cards are chosen at random. In reality a player chooses four cards from the six they are dealt, so you would actually need to calculate the odds of getting a six-card hand of zero points (about 1.36%), then getting a starter card that means you still end up on zero points. This comes out at approximately 0.43%, assuming the four cards you keep from the initial six are chosen randomly.