# What are the odds for a “19” (zero pt) hand in Cribbage?

I am doing a math paper for school using Cribbage. I know how to use factorials to calculate the odds of getting the perfect 29 hand.

My question is, how many 5 card hands (4 in hand and turn up card) add up to zero?

• Maybe it’s worth showing what’s you’ve tried so far? Whilst it seems a valid question for the site I wouldn’t feel comfortable doing someone’s school work for them when they should be making every effort themselves and the teacher being able to show how to improve it. – StartPlayer May 3 '20 at 7:20
• Please add details that will allow someone that do not know Cribbage but knows probability (or just combinatorics) could help you – Cohensius May 3 '20 at 11:55
• I can use C(52,6) to get the total number of hands possible with a 6 card deal. Certain sites say that the probability of getting a 0 pt hand is about 7%. A 0 pt hand implies 5 cards with no pairs, no runs, no cards adding up to 15. I can calculate this manually but am curious about the mathematical equation. – Seattle May 4 '20 at 12:29
• Take any one card. Calculate the number of cards you should NOT draw next. Put the different numbers (depending on the card drawn) in a (wheighted) tree. Repeat for all cards. You might see a patern at some point and do a mathematical induction, but I would not count on it. – RandomGuy May 4 '20 at 13:16
• This is a significantly more difficult computation than computing the probability of a 29. There are a ton of edge cases to work out. If you are not familiar with inclusion-exclusion in combinatorics, you may want to look that up. – John May 5 '20 at 15:24

## 1 Answer

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.