Mind-Blowing Math
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wheels in the world, spinning 24/7,
a sequence of the numbers 1-36, in
correct order, would occur once every
20 million years.
Somebody should have the answer for you pretty soon, since the number of possible permutations with 52 is known.
52!=80 658 175 170 943 878 571 660 636 856 403 766 975 289 505 440 883 277 824 000 000 000 000 says my calculator, but is that the number we want?
not far from 8 * 10^67 I believe, if it is what we want [I shouldnt be doing this but what the ...]
"The Milky Way contains between 200 and 400 billion stars" per wikipedia article. Let's use 4 * 10^11 [we'll only use the powers of 10 anyway]
4.32 × 10^17 = seconds in the universe; that times 10^3 and we are in the 10^20 zone
clearly you don't use the factorial for all those trillions this and that and not expect it to exceed 52! LOL. You multiply them?
adding the powers, that's 11+9+9+9+20=58, so that gives us 10^58 as opposed to 10^67 . There you are, confirmation probably via the wrong procedure giving the wrong answer LOL.
Quote: MindOverMatterOdiousgambit, I applaud your quick response (especially so late at night or early in the morning, relatively speaking) referencing the 52! factorial. The article I was reading can be found here: http://www.cracked.com/article_22432_6-bizarre-statistics-that-prove-math-black-magic.html Have you ever dabbled in computer programming? I sense something familiar in the way that coders think of things in terms of orders of complexity with how you are "eye-balling" vastly large numbers. Thank you for bringing some hard figures into the discussion.
well, it's nice to see at least part of what I wrote was right. And today, you have calculators available that just kick this stuff out whether you know what you are doing or not, sort of like being handed an AK-47 while all you actually know about guns is that time you shot a .22 pistol LOL.
in school, I wasn't too bad in applying what math I knew, but by the time I was in college so many of the other students were so far ahead of me I was patently too remedial to even think of taking the advanced courses in math. At the time, I came away feeling that anybody who would need to do well in math in college would need to have as a minimum the equivalent of 2 years college math under his belt before he stepped foot on campus ... math that he excelled in too. I don't know if that has all changed or what it is like now.
so, no, computer programming was not on the agenda, although I learned some basic elementary ideas about it.
Quote: MindOverMatterI read something today that seemed very hard to reconcile with common sense, but I found it to be fascinating: When you shuffle a deck of 52 cards, there is almost a mathematical certainty that not only are you creating a sequence of cards that has never before existed in the history of the world, but that the sequence you just created will never again be duplicated. Here is a quote from the article: "It has been calculated that if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people has a trillion packs of cards and somehow they manage to make unique shuffles 1,000 times per second, and they'd been doing that since the Big Bang, they'd only just now be starting to repeat shuffles." If anyone would like to challenge this claim, I'd be interested in your thoughts.
The only problem with this statement is, it assumes that all shuffles are perfectly random - that is, any particular card has a 1/52 chance of being the first card, a 1/52 chance of being the second, and so on.
In fact, if you take a 52-card deck and shuffle it "perfectly" (i.e. divide it into two 26-card piles and "riffle shuffle" them starting with the "bottom 26" pile) eight times, the cards end up in the same order in which they started.
However, your point is valid - large numbers can get to the point of losing all meaning surprisingly quickly.
Quote: ThatDonGuyThe only problem with this statement is, it assumes that all shuffles are perfectly random - that is, any particular card has a 1/52 chance of being the first card, a 1/52 chance of being the second, and so on.
In fact, if you take a 52-card deck and shuffle it "perfectly" (i.e. divide it into two 26-card piles and "riffle shuffle" them starting with the "bottom 26" pile) eight times, the cards end up in the same order in which they started.
Well, what they mean by a perfect shuffle each time is a perfectly randomly rearranged deck each time
Quote: odiousgambit
52!=80 658 175 170 943 878 571 660 636 856 403 766 975 289 505 440 883 277 824 000 000 000 000 says my calculator, but is that the number we want?
That's 80 unvigintillion, just for the record.
Quote: odiousgambitadding the powers, that's 11+9+9+9+20=58, so that gives us 10^58 as opposed to 10^67 . There you are, confirmation probably via the wrong procedure giving the wrong answer LOL.
Odious,
Your maths is correct but 1 trillion is 10^12 not 10^9. so it works out as 11+12+12+12+2067 which matches the articles claim.
If you had a different order of the cards for every second (or 1000 per second) since "the big bang" then yeah, there'd be no duplicates so far.
But if you randomly shuffle the cards 1000/second, then there is no mathematical certainty there hasn't been a duplicate.
You remember it takes 22 people in a room to be >50% sure two have the same birthday (essentially look at 1 - 364/365 * 363/365 etc.
If we try that some idea on 1000 (perhaps there's a 1000-recipe book and what's the chance two people made the same dish at a party) then the answer is 37 (using a simple excel spreadsheet).
Number of people needed so the chance of two picking the same (from the number of possibilities) now exceed 50%..
| Perms | Number needed |
|---|---|
| 365 | 22 |
| 1000 | 37 |
| 10000 | 118 |
| 100000 | 372 |
| 1000000 | 1177 |
Thus it looks to be about SQRT(possibilities) - so if there were 10^34 decks dealt, there would be a 50% chance of two being the same. Still unlikely but not quite as large as 8E+67.
Quote: charliepatrickYou remember it takes 22 people in a room to be >50% sure two have the same birthday
It takes 23.
With 22 the probability of a common birthday is 47.57%. With 23 it is 50.73%.
Before somebody corrects me, this does not consider leap days.
Question: The probability of winning the PowerBall lottery is 1 in 175,223,510. What is the least number of randomly chosen tickets for their to be a greater than 50% chance of a common ticket?
Quote: bigfoot66I read the article a couple of days ago, it is poorly written and I thought about emailing the author but thought better of it. He was trying to explain the boy or girl paradox http://en.wikipedia.org/wiki/Boy_or_Girl_paradox that we have been discussing in the 2 coins threads ad nauseum and stated that if I guy tells you he has a sibling, then there is a 2/3 chance that the sibling is female. This, of course, is not accurate. the sibling is 50/50 to be male or female.
Read:
Quote: wikiA shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male?
Quote: MindOverMatterI read something today that seemed very hard to reconcile with common sense, but I found it to be fascinating: When you shuffle a deck of 52 cards, there is almost a mathematical certainty that not only are you creating a sequence of cards that has never before existed in the history of the world, but that the sequence you just created will never again be duplicated. Here is a quote from the article: "It has been calculated that if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people has a trillion packs of cards and somehow they manage to make unique shuffles 1,000 times per second, and they'd been doing that since the Big Bang, they'd only just now be starting to repeat shuffles." If anyone would like to challenge this claim, I'd be interested in your thoughts. If anyone would like to share any other fun facts about the mathematical side of gambling that seem to be too crazy to be true, please share those as well. One of the things that makes gambling so interesting to me (I'm mostly a blackjack player) is that the experience combines the fascinating yet rather sterile world of pure mathematics/statistics/logic along with the element of human psychology and the emotional roller coaster. Thanks for your replies.
Here's a calculation.
It is estimated that the Milky Way Galaxy has 100 billion stars. That's 10^11.
A trillion planets with a trillion people on them with a trillion decks of cards is 10^36.
11 plus 36 plus 3 for the thousand shuffles per second is 10^50.
It has been about 13.8 billion years since the Big Bang, and if you figure 31,557,000 seconds per year (including a quarter of a day per year for leap years,) that adds up to 435,486,600,000,000,000 seconds since the Big Bang, or for those of you who prefer scientific notation, 4.355 * 10^17.
That results in the total number of shuffles since the Big Bang as 4.355 * 10^67.
A deck of cards has 8.0 * 10^67 possible permutations.
So any given permutation's expected value for number of appearances in this simulation is not too far above 1/2.
Quote: WizardIt takes 23.
With 22 the probability of a common birthday is 47.57%. With 23 it is 50.73%.
Before somebody corrects me, this does not consider leap days.
Question: The probability of winning the PowerBall lottery is 1 in 175,223,510. What is the least number of randomly chosen tickets for their to be a greater than 50% chance of a common ticket?
