Is it 1 in 46,656?

November 29, 2010 10:01 AM

Probability filter: after eating the TD turkey we play the Turkey game which consists of tossing six dice. The six faces of each die are carved with the letters that spells turkey. Each combination of letters earn a different score (for example T U is 5 points, 3 Ts wipe all the points earned, etc.) with the first TURKEY being the winner. How many tosses would you need to spell TURKEY?

You would not believe how many different answer a bunch of supposedly intelligent people gave, from 6! to 6 to the 6th power. Which answer is right? Explain to me how and why, please.

posted by francesca too to education (15 answers total)

Let's consider the dice as if they are rolled in sequence - it doesn't make a difference in the probabilities.

The first die can be anything, and be consistent with getting TURKEY.

The second die can be anything besides the first die's result.

The third can be anything besides the first two, and so on.

So, the probability of TURKEY is 1 * (5/6) * (4/6) * (3/6) * (2/6) * (1/6).

To find how many rolls we'd expect, divide 1 by that.

1 / [ 1 * (5/6) * (4/6) * (3/6) * (2/6) * (1/6) ] = 64.8 rolls.
posted by Earl the Polliwog at 10:05 AM on November 29, 2010 [2 favorites]

There are 6^6 possible combinations of dice throws, and 6! possible combinations that give all the letters of TURKEY.

So, 720 out of 46,656 possible combinations, or about a 1 in 65 chance of throwing TURKEY provided you don't care in which order the letters come up.
posted by The Michael The at 10:08 AM on November 29, 2010 [1 favorite]

How many tosses would you need to spell TURKEY?

This is kind of a vague question - theoretically, you could roll 6 dice 1 million times and never spell "TURKEY", although it's unlikely.

Given that you are rolling all six dice at once, the probability of rolling 6 unique die faces is

(5/6 * 4/6 * 3/6 * 2/6 * 1/6) or 120/7776 or about 1.5%.

On preview, what Polliwog said.
posted by muddgirl at 10:08 AM on November 29, 2010 [1 favorite]

Given that you are rolling all six dice at once

Deletion error! I meant to say that, given you are not doing some Yatzee thing which allows re-rolls...
posted by muddgirl at 10:09 AM on November 29, 2010

Maybe a little pedantic, but ^6!⁄_6⁶ is the probability of drawing TURKEY on any one sample, and the reciprocal is the expected number of samples until TURKEY is drawn.
posted by Estragon at 10:12 AM on November 29, 2010 [2 favorites]

Very fast indeed, and I even understand the answers! (which amazes me)
posted by francesca too at 10:17 AM on November 29, 2010

That's some nice sup/sub HTML there Estragon!
posted by nicwolff at 11:36 AM on November 29, 2010

Really the question isn't the probability of rolling it on one roll, but how many you'd need to roll to guarantee it. I assume that if the odds are 1/65 on each roll, that if you roll 65 times, then the % chance of rolling TURKEY at least once is still a high number, but still has to be less than 100%. How do you figure that percentage out?
posted by empath at 1:11 PM on November 29, 2010

empath: I think that 65 rolls is the point at which there is a ~50% chance of having rolled TURKEY. It is the expected number of rolls, and thus the peak of the distribution. I think it will actually be slightly less than 50%, because you have an infinitely long tail in the "more than 65 rolls" direction, while you can't ever roll TURKEY in less than one roll.

And of course there is no number of rolls that will guarantee you roll TURKEY.
posted by 256 at 1:20 PM on November 29, 2010

empath:

The probability that you won't roll a turkey is (1-120/7776). The probability that you won't roll any TURKEYS in 36 rolls is (1-120/7776)^36 or 57%. So the probability that you will roll at least 1 turkey in 36 rolls is [1 - % you don't roll any turkeys], or 42%.
posted by muddgirl at 1:22 PM on November 29, 2010

Whoops, I don't know where 36 came from. I get 64% for 65 rolls.
posted by muddgirl at 1:27 PM on November 29, 2010

If it's only 64% then that's hardly a guarantee, that's almost a coin flip. How many times would you have to roll to be 95% certain you'd get turkey?
posted by empath at 1:45 PM on November 29, 2010

I couldn't get my math to work out analytically but according to Excel it will take 193 rolls to be 95% certain.
posted by muddgirl at 2:33 PM on November 29, 2010

Remember that this is just the probability of getting at least 1 TURKEY in that timeframe. The position of the TURKEY is not calculated - it is equally likely that you will roll the TURKEY on the first roll as the last.

If you want to know the probability that you will roll 192 no-TURKEYS followed by a TURKEY to end your rolling, that will be a much much lower probability.
posted by muddgirl at 2:38 PM on November 29, 2010

This actually sounds like a fun game - mind posting the rest of the combinations and rules?
posted by chrisinseoul at 9:32 AM on November 30, 2010