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?

  • 3
    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
  • 1
    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
  • 1
    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

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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.