Question for the math guru's

[Britman]

5 dice, each with 6 sides.

What's the odds of rolling the 5 dice and getting the same number?

6x6x6x6x6?

[sirepair]

What is the probability, when tossing six dice, of throwing each number once i.e. achieving equipartition?

The total number of ways equipartition can occur is 6! The first die can clearly be any of the six numbers. the second die one of the five remaining, and so on, giving a total number of ways of 6 x 5 x 4 x 3 x 2 x 1 = 720.

The number of possible outcomes is power(6,6) since there are six ways each of the six dice can fall. Therefore this is equal to 46656.

So the probability of equipartition is:-

720/46656 = 0.0154 or 1.5432%

Most people would be surprised to discover that if you threw six dice, on about 98.5% of occasions, at least one number will appear more than once.

I stayed in a Holiday Inn Express last night....
(actually from: http://www.peterwebb.co.uk/probability.htm#Combinations%20and%20permutations)

[Mellow]

It's 5 dice... so, 5 to the power of 6 right?

[Craig]

According to my quick and usually inaccurate calculations ....
the odds of having 5 dice roll the same number are 1 in 7776 times.

[st1300r]

1st die doesn't matter it can be anything.
2,3,4,5
1/6*1/6*1/6*1/6
1/1296 That's my guess. :)

Now
Why is it called a combination lock?

[dreddd2000]

I get a 1 in 30 chance of that role combination, but I'm not certain I'm using the right math.

Odds of getting one die to roll a 6 is 1 in 6. Odds for getting the second die to roll a 6 is still 1 in 6. Carry that logic forward and I end up at a 1 in 30 chance.

[pretbek]

Britman,
What does the casino call this game and how much did you bet on it?

[st1300r]

According to my quick and usually inaccurate calculations ....
the odds of having 5 dice roll the same number are 1 in 7776 times.

That would be if all 5 were one particular number.
If they just had to be all the same arbitrary number I think its 1/1296.
:confused:

[Blrfl]

The probability of a six-sided die rolling any given value is one in six. (P = 1/6)

The probability of n events happening together is the product of the probabilities of each of the events. (P = P1 x P2 x ... x Pn) So for any five dice, the probability of any single outcome (i.e, all ones but not all twos through sixes) is 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/7776.

IMPORTANT NOTE: This assumes that if you're looking for an outcome of, say 1/2/3/4/5, you don't consider getting 5/4/3/2/1 the same, even though you can rearrange the dice to get the same result. But since we're looking for an outcome of all the same number, this doesn't come into play.

There are six instances where you'll get all the same digits, so for that you take the sum of the probability of each of those six outcomes, so P = 1/7776 + 1/7776 + 1/7776 + 1/7776 + 1/7776 + 1/7776 or 6/7776.

Reduce that a bit and you get P=1/1276 or 0.000771605 or about 0.07%. I'd categorize that as "pretty slim."

--Mark

[Kevin_56]

What BLRFL said. It has been a few years since Probability and Statistics, but you are right by my book.

Kevin

The probability of a six-sided die rolling any given value is one in six. (P = 1/6)

The probability of n events happening together is the product of the probabilities of each of the events. (P = P1 x P2 x ... x Pn) So for any five dice, the probability of any single outcome (i.e, all ones but not all twos through sixes) is 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/7776.

IMPORTANT NOTE: This assumes that if you're looking for an outcome of, say 1/2/3/4/5, you don't consider getting 5/4/3/2/1 the same, even though you can rearrange the dice to get the same result. But since we're looking for an outcome of all the same number, this doesn't come into play.

There are six instances where you'll get all the same digits, so for that you take the sum of the probability of each of those six outcomes, so P = 1/7776 + 1/7776 + 1/7776 + 1/7776 + 1/7776 + 1/7776 or 6/7776.

Reduce that a bit and you get P=1/1276 or 0.000771605 or about 0.07%. I'd categorize that as "pretty slim."

--Mark

[Mark]

+1 Kevin,

I had just found the relevant section in my Stat book and when I went to start typing the "1/6 x 1/6" stuff I found Mark had done an excellent job.

Mark

[Britman]

Britman,
What does the casino call this game and how much did you bet on it?

My local bar started a $1 roll of the 5 dice. If you lose the money goes in the large pickle jar. No-one had won since March.

My buddy Sean rolled 5 5's last night.

The contents of the 2 jars was $1,785. Pretty good considering he was in the parking lot deciding whether to come in for a quick beer on his way home.

[Mark]

Reduce that a bit and you get P=1/1276



$1,785.


So at $1 per throw this took quite a bit longer than the predicted 1276 throws.

[Britman]

So at $1 per throw this took quite a bit longer than the predicted 1276 throws.

Surely every time you throw the odds remain the same. They don't reduce as throws accumulate?

[Scooterb]

So at $1 per throw this took quite a bit longer than the predicted 1276 throws.

But not abnormally so...I bet that Britman's friend was happy that it took longer!

[Blrfl]

Each throw's odds are independent of the others, and there's no guarantee that you'll hit it in 1276 throws. It just means that on average, you'll get one in 1276.

--Mark

[MLRickards]

Wow, a sure sign that I've lived in Denton County too long. I thought the title of this thread read "Question for the meth guru's". Thought maybe there was a new theory on meth-based riding or something.

Sorry, back to the statistical discussion...

[Mark]

Surely every time you throw the odds remain the same. They don't reduce as throws accumulate?

You are absolutely right.
But it took almost 30% longer than the strict probability calculation predicted.

Mark

[George]

Isn't .07% statistically close to zero?

[Trekker]

I would have said infinity, then the treadmill under the bar would break, causing the earth to shift in orbit negating the imminent global warming crisis.....

or... I could have been hitting the meth today! :hyp1:

Hypothetically speaking of course.

[Blrfl]

Isn't .07% statistically clost to zero?

That depends on what you're looking at.

An airline that flies a million passengers a month and crashes two planeloads of 350 each month, every passenger has a 0.07% chance of being in a crash. No airline is going to want that, no matter how small a fraction it is.

--Mark

[George]

OTOH, buying _2_ super lotto tickets doesn't really double yur chances of winning, either: 2 x 0 = 0. :D

[XLR8]

OTOH, buying _2_ super lotto tickets doesn't really double yur chances of winning, either: 2 x 0 = 0. :D

That's true, but 2 x .0001 = .0002. Seems to me, like that increases your chances.

[Blrfl]

If you play two different numbers, your chances are twice what they were. (Still infinitesimal.)

If you play multiple tickets, you can maximize your winnings in the event you hit the jackpot by playing the same number. If you play four tickets that win and someone else plays a single ticket with the same number, you'll end up with 80% of the pot instead of 50% had you only played a single ticket.

--Mark

[XLR8]

If you play two different numbers, your chances are twice what they were. (Still infinitesimal.)

If you play multiple tickets, you can maximize your winnings in the event you hit the jackpot by playing the same number. If you play four tickets that win and someone else plays a single ticket with the same number, you'll end up with 80% of the pot instead of 50% had you only played a single ticket.

--Mark

Very true, but if the pot's 341 million bucks, do you even care if you got 50%, or 80%? I don't think it would matter to me. If I bought 4 tickets, I'd want to get 4 different chances.

[DAS]

Why when I started reading this thread I knew Blrfl would be the first to give the right answer?

[Trekker]

Why when I started reading this thread I knew Blrfl would be the first to give the right answer?

I think on the bottom of his business cards it says- "Super Genius, Cranium Maximus"

.... funny, he don't look like no coyote!

[Blrfl]

Why when I started reading this thread I knew Blrfl would be the first to give the right answer?
I think on the bottom of his business cards it says- "Super Genius, Cranium Maximus"

Just giving the ol' noggin a workout...

--Mark