What are the chances of spinning a Yahtzee on the first roll?

August 17, 2009 6:00 PM

What are the chances of spinning a Yahtzee on the first roll?

This happened during a real game and we wanted to find out what the odds were of it happening. (A Yahtzee is 5 dice of the same number.)

posted by R. Mutt to grab bag (21 comments total)

⅙^5 = .0128%
posted by floam at 6:01 PM on August 17

floam's answer is the probability of a particular number coming up as Yahtzee; for any Yahztee, it's (1/6)^4, the probability that the other four dice match the first one.
posted by ldenneau at 6:03 PM on August 17 [1 favorite has favorites]

According to Wikipedia, it's 1 in 1296.
posted by MaryDellamorte at 6:03 PM on August 17

Wait, no. Sorry. Totally wrong.
posted by floam at 6:04 PM on August 17

Yes, that was the exact discussion we non-math folk were having.
posted by R. Mutt at 6:06 PM on August 17

⅙⁴ = 1 in 1296 in case you were wondering.
posted by @troy at 6:09 PM on August 17

Idenneau is correct. It's equivalent to just take the probability of one of the possible 6 yahzees (which is what I accidentally did), and multiply it by six, if that makes it sink in better.
posted by floam at 6:09 PM on August 17

floam's last post is right -- but if you need the (^4) to make sense, think of it as you get to roll one die and have it be random, and then you need the other 4 to match it -- and each die has a 1 in 6 chance of that.
posted by brainmouse at 6:19 PM on August 17

Unless I am mistaken (not being a math person) it is same same odds as on the second roll. And the third roll. etc.

I'm not saying. I'm just sayin'.
posted by SLC Mom at 6:40 PM on August 17

Not necessarily, SLC. If you've kept matching dice on the first roll, then your odds are increased based on how many matching dice you already have.

And, of course, if you've kept non-matching dice on the first roll, the chance of getting Yahtzee on the second roll is nil.
posted by jacquilynne at 7:05 PM on August 17

SLC Mom makes me think: if you save 1 of the dice, your odds of getting a yahtzee are the same as if you save none of the dice -- regardless of which roll it is. If you save more than one die (of the same #), obviously, your odds go up...
posted by brainmouse at 7:07 PM on August 17

um... I'll beg to differ with all the numbers provided so far, but only for this reason: you asked what the probability was to do this on the first roll. What has been provided is the probability of throwing a Yahtzee, not a Yahtzee on the first roll of the game.

So, given that a game of Yahtzee consists of filling out 13 categories within 3 rolls for each category, each roll also must consider its exact order. We are looking to ensure that one Yahtzee occurs within a set order, namely the first position. To simplify the math, lets assume that only this Yahtzee will the participant be throwing only once throw - meaning that we will roll 12 more categories of 3 rolls of which we do not care the outcome. (12x3)+1 = the game lasts 37 rolls: meaning to do it precisely we now multiply 1 in 1296 by the probability we re in position one: which means our odds are 1 in 47,952.

Yes, its just as likely to throw a natural Yahtzee at any given roll; however, when forced into the first roll only...
posted by Nanukthedog at 7:15 PM on August 17

Nanuk -- you are not right. You can't say "Yes, its just as likely to throw a natural Yahtzee at any given roll; however..." anything. The odds of rolling a Yahtzee when you are rolling all 5 dice is always the same: 1 in 1296. It doesn't matter which roll this is, or what turn it is, or what else is going on. Whenever you are rolling all 5 dice -- which includes the first roll of the dice -- the odds are 1 in 1296.
posted by brainmouse at 7:21 PM on August 17 [2 favorites has favorites]

Nanuk: You've just given the probability of playing a game in which a natural Yahtzee is thrown exacly once.
posted by mr_roboto at 7:46 PM on August 17

My teacher would have written it 1 in (6^5)/6, to clearly illustrate the thought process behind his solution.

Can I leave my wanker hat on for an extra moment and ask was it a trick question? One doesn't "spin" a Yahtzee.
posted by uncanny hengeman at 7:57 PM on August 17

Brainmouse: yes, the odds of rolling a Yahtzee at any one time are 1 in 1296. A pure game lasts 13 rounds of 3 spins, meaning that a person will make a maximum of 39 spins, by definition in this case we'll lower that to 37. If I am not concerned with when the vent occurs, and I am only concerned with the event of a natural, when picking up the dice for a natural Yahtzee to occur in the fifth round you are correct: 1 in 1296. for a natural Yahtzee to occur on the second round, you are correct: 1 in 1296. Reread the question very strictly: What are the chances of getting a Yahtzee on the first roll? Yes, your chance of getting a Yahtzee on a natural roll is 1 in 1296, but for your first roll of the game to be a Yahtzee you are examining ordinal probability as well. It is no longer 5 equal dice in the context of five dice, it is five specific dice out of one hundred and eighty five dice which will be thrown.

Now Mr_roboto does make a very strong point, this assumes a natural Yahtzee is thrown exactly once...
posted by Nanukthedog at 8:22 PM on August 17

You're just really, really wrong here, Nanuk. Look at it this way. The chance of picking up five dice and throwing them and getting all the same number is 1 in 1296, as you agree.

Now, let's assume it's just happened. Amazing! A natural Yahtzee right off the bat! What effect do all the further 180 dice have on that such that they would change the probability?

Now, assume that is just didn't happen. You rolled a 1, two fours and two 6s. Boo. No Yahtzee for you. What different do the next 37 rolls make to whether you got a natural on your first roll?

None. None at all.

You're basically assuming you can get at most one yahtzee in a game, and what's the chance that a) you'll get it and b) it'll be on the first roll. That's not the question.
posted by jacquilynne at 8:31 PM on August 17

Nanuk:

I think I know what you're saying. You're asking: "Out of some finite but immense number of Yahtzee games, how many would open with a Yahtzee on the very first roll?" And the answer, given enough games, will approach: 1 in 1296.

Remember: The first player lifts the cup to roll the dice. If she rolls a Yahtzee, we'll have a Yahtzee-opening game. If not, we won't. How often will she do it? One out of 1296 games.
posted by argybarg at 8:32 PM on August 17

I was completly forgetting about the actual playing of the game, keeping back one or two good dice to try to match. I was only thinking about rolling all 5 dice at once. Any time you do that, that chances are the same as any other time.
posted by SLC Mom at 10:14 PM on August 17

Nanuk is wrong, but not for the reasons suggested above. This is because there's some minor ambiguity in the question. It can be read two ways:

A) Assuming a Yahtzee will be rolled, what are the odds of it happening on the first roll?

B) What are the odds of rolling a Yahtzee?

The answer to "A" is 1 in 37 because a "pure" game consists of 37 throws and we have assumed a Yahtzee will be thrown at some point. Thus, Nanuk is incorrect because he started out assuming a Yahtzee would be thrown, but then went ahead and multiplied by the odds of throwing a Yahtzee anyway.

Obviously, the answer to "B" is 1 in 1296.
posted by aheckler at 5:00 AM on August 18

Reread the question very strictly: What are the chances of getting a Yahtzee on the first roll? Yes, your chance of getting a Yahtzee on a natural roll is 1 in 1296, but for your first roll of the game to be a Yahtzee you are examining ordinal probability as well.

Nanukthedog, imagine a modified version of Yahtzee, played with 2 coins. Instead of all the different Yahtzee categories, there are only two: Match (both coins are heads or both coins are tails) and Nonmatch (one head and one tail). There are only 2 rounds. However, each person gets 1,000 flips each round, that is to say, she can flip and then reflip any of the coins she does not want to keep, up to 1,000 times per round. What is the probability that someone will flip a Match on the first flip of coin Yahtzee?

There are 4 outcomes when you flip two coins: HH, TT, TH, HT. Two of them are matches, so 2/4 = 1/2. A pure game lasts 2 rounds of 1,000 flips meaning that a person can make a maximum of 2,000 flips, but in this case we'll lower it to 1,001 flips. 1/2 * 1/1,001 = 1/2,002, right?

Go play coin Yahtzee like 50 times. If the odds were really 1/2002 of opening with a Match, you probably won't see one right off the bat, right? I think you'll find you see an opening Match alot, because the probability is 1/2, just like the probability of a opening Yahtzee using dice is 1/1296.
posted by 23skidoo at 8:25 AM on August 18