Ask MetaFilter posts tagged with probability

Probability question (in need of code)

chrisamiller — Wed, 13 Aug 2008 22:04:42 -0800

It's late, I'm tired, and I have a probability and coding question that's fairly simple. Say I have 5 buckets (A,B,C,D,E), with different colored balls in them. The probability of removing a red ball is different for each bucket:

A: 1/3
B: 1/4
C: 1/16
D: 1/6
E: 1/9

If I draw 1 ball from each bucket, what is the probability (P) of drawing at least N red balls?

Doing it for small numbers of buckets is easy enough, but I have thousands of buckets here. Given that I know the probabilities for each bucket, what's the easiest way to calculate P? I suspect that there's an R function that makes this a breeze, but I'm having trouble tracking it down.

I can do basic operations in R, and I'm also open to functions or pseudo code from your favorite language. Ruby and Perl preferred, but I can use others if they get the job done.

For good measure, here's a preemptive "This isn't homework-filter"

How to calculate probability of being dealt certain cards?

b_thinky — Wed, 09 Jul 2008 17:44:11 -0800

You're dealt 8 cards face down from a 52 card deck. All values are determined by the number on the card, not the suit. What is the probability of each card being dealt? The first card you're dealt has a 4/52 chance of being any of the 13 values (2 thru Ace). The 2nd card you're dealt has a 4/51 chance of being any of the remaining 12 values, plus a 3/51 chance of being the same value as the first card.

This is where my retard brain ceases to work. How do I know the odds of the 3rd card, given that the 2nd card may or may not be the same as the 1st card? If the 2nd card has a new value, the third card is 4/50 of the 11 remaining cards, plus 3/50 it's the same value as the 1st card and 3/50 it's the same as the 2nd card. But if the 2nd card is a repeat of the 1st value, there is a 4/50 chance of it being one of the 12 remaining values and a 2/50 chance of it being the same as the 1st and 2nd card.

So you see the dilemma I have calculating the odds for the 3rd card to be dealt until the 8th. I know I'm probably looking at this entirely the wrong way but I've always been a math retard beyond basic arithmetic. Anyone care to explain how this works?

21 judges give 4 contestants a 4, 3, 2, or 1. How often is there a first place tie?

23skidoo — Tue, 24 Jun 2008 11:22:02 -0800

Imagine a contest, with 21 judges and 4 contestants. Judges must give one contestant 4 points, one contestant 3 points, one contestant 2 points, and one contestant 1 point. The winner is decided by summing the scores of all 21 judges. Mathematically speaking, what percentage of the time would you expect there to be a tie for first place after all 21 scores have been added?

Oh, boy, parallel universe #57339! That's where I'm a Viking.

Rhaomi — Sat, 24 May 2008 17:39:43 -0800

How would the many-worlds interpretation work on the human level, if at all? (I'm no quantum physicist, so please forgive me if the following is woefully simplistic, ridiculously naive, and/or hopelessly wrong)

The way I've heard it explained, the many-worlds interpretation of quantum mechanics holds that for every situation in which multiple outcomes are possible, each one of those outcomes does happen -- albeit in its own universe. That the universe we perceive is just part of an inconceivably large multiverse of infinitely branching possibilities, and that every interaction between every atom everywhere in the universe creates another one, or multiple ones, all the time.

I've also heard that because every possible outcome occurs, even the most bizarrely improbable event has happened in at least one universe. This makes sense if the ideas in the above paragraph are true.

For instance, it is incredibly unlikely that a fair coin could come up heads twenty times in a row -- the odds are about 1,048,576 to 1. But if each coin toss branches into a universe where it lands heads and a universe where it lands tails, then at the end of the line one of the million+ universes would see it land heads all twenty times. Of course, most of the rest of the million branches would see a mixed outcome, so from the point of view of a single universe the odds are still very unlikely. But the many-worlds theory says it does happen somewhere.

But when you think about it, it wouldn't be that simple. For starters, each coin toss would make more than two universes. A lot more. For example, a single coin toss could have two universes where the coin lands tails, but one universe sees it land one centimeter further to the right than the other. And there would be a small set of worlds where the coin landed perfectly on its side. And an even smaller minority where all the molecules of the coin spontaneously evaporated at the same time. A colossally improbable event, but possible.

And hey, if this holds true for molecules and small objects like coins, could you not also extend it to the rest of the world (which is just a collection of 10^huge molecules)? Must there be a universe out there where, say, every person who bought a Florida State lottery ticket happened to pick the same number, which was the winning one? Or where every building on Earth suffered simultaneous structural failure? Or where everyone spontaneously decided to break into Broadway-style song and dance? And a trillion variations on these and other scenarios, each slightly different from the other? And that the only reason we (most of our selves?) don't experience these things is because the infinity of "normal" universes where probable things happen outnumbers the infinity of universes where "impossible" things happen?

It feels absurd, like I'm talking about the Infinite Improbability Drive instead of a theory of physics, but I'm not seeing why it shouldn't be true. I've tried finding answers, but most of the literature out there (with the occasional exception) deals with quantum physics in a dry, academic context that limits the discussion to the atomic level. And of the material that imagines crazy outlier universes like the ones I described, I don't have a good way of telling if the physics involved is real or just taking artistic liberties for the sake of interesting fiction.

Must I mourn for the Earth somewhere out there that suffered Total Existence Failure? (And the one that suffered it five minutes later, and the other one that suffered it sixteen years later, and the one that suffered it partially, losing the western hemisphere, and the one that lost the eastern hemisphere, and the one that suddenly split into two planets, and the one where Australia turned into gelatin, and etc.)

Oh, and I know that we still aren't sure about which interpretation of quantum mechanics is correct, and that even if we knew the many-worlds theory were right, we'd have no way to observe other universes. I just want to know if the things I described are allowable in the context of the theory as it's understood today.

What is a good sample size for determining whether dice are truly random?

bugbread — Wed, 30 Apr 2008 11:41:41 -0800

How many times would I have to roll a standard 6-sided die to get a statistically representative view of whether it was truly random or not? I have a bunch of dice that I haven't used in years. The other day, I was playing with one, and I noticed that the 5 came up fairly often. I started rolling the die and writing down the results, to see if it was just a short term statistical fluke, or observer bias, or if the die really favored the 5.

I think I ended up rolling it around 200 times, and 5 definitely had a significant edge.

Now, I'd like to check my other dice. However, I'm not a statistician. I understand that, obviously, the larger your data set (the more times you roll each die and write down the results), the better your analysis of non-randomness will be. However, I know that it isn't necessary to roll the die 1 billion times to check for randomness, that there is some generally accepted statistical minimum, below which the margin of error is too large, and above which the margin of error is generally considered acceptable.

How many die rolls is that point?

Probability and Truth

binturong — Sat, 12 Apr 2008 12:59:35 -0800

Suppose you take a test for a rare type of cancer that affects 0.01 percent of the population. The test is 98 percent reliable. You get a positive reading. What are the chances you have the cancer? I read this probability puzzle today and the writer said the statistical chances of you having the cancer in this scenario are less than half a percent. I don't get it. Isn't the rarity factor irrelevant compared with the test reliability? Please explain.

Do you know my friend?

Gungho — Tue, 01 Apr 2008 13:37:58 -0800

Mathematics filter: What are the odds of a pair of my friends meeting another couple who are also my friends, but they have never met before? Bonus. This happened on vacation in another city where neither of them, nor I, have ever lived... Is this an example of six degrees of separation?

what's the probability distribution, kenneth?

russm — Fri, 18 Jan 2008 19:20:25 -0800

for the statisticians: what probability distribution do I want to fit to my data, and how? I have a collection of widgets. they age, and as they age some of them get tweaked. it's been suggested to me that given enough time every widget will eventually be tweaked, but I'm not convinced. the data I have covers ~64000 widgets, and for each widget is either the age at which it was tweaked, or alternatively the current age and the fact that it has never (yet) been tweaked.

as I understand it (I'm a database programmer with an interest in maths but very little working stats) I want to fit some probability distribution to my data and see whether the area under the probability mass function is < 1.

the 2 things that make this different from everything I've been able to find on the 'net are

1) I specifically don't expect the cumulative distribution to asymptote to 1, which appears to count out obvious candidates like a Poisson distribution

2) my data contains negative examples - those widgets which have not yet been tweaked, and may never be. I'm assuming these are relevant to the shape of the final distribution, but I can't work out how to get tools like R to take them into account.

the best advice I've had so far is from an engineer friend who said to chuck the data on only the tweaked widgets into R, fit a Poisson distribution and be done with it. unfortunately this fails on both of the above points. (then again, all he normally cares about is failure rates in aircraft parts, and you know that given enough time every lug will eventually break).

so, if there are any stats geeks who understand what I'm trying to do and can point me at tools/docs/info on how to do it you'd rock my world.

complex dice probability make head esplode

DetonatedManiac — Fri, 21 Dec 2007 13:52:56 -0800

I need a tool to help me analyze the probability of certain events occurring regarding a pool of X dice for a game I am creating. See my examples below. I am looking for a dice statistics program or, baring that, a set of formulas I could input into excel, that can answer the following questions. Graphing capabilities are a plus.

Some of the events I want to analyze:

1) The probability that, from a pool of X dice, Y dice can be chosen by a player such that the sum of the Y dice is equal or lesser than a number N.

2) Related to the previous question; of the pools that succeed what is the distribution of how close they are to hitting N, (for instance, if the sum of Y dice is 10 and the number the player was aiming for was 12 then they are only off by 2).

3) In a pool of X dice How likely is it that there will be Z sets of doubles (say two 6s or snake eyes) and, more specifically, what are the chances of getting a specific set of doubles (say just snake eyes).

I am pretty statistics theory inept (never took it in school) but I am average to above average with math in general... basically please keep any advice in laymen terms as much as possible. Thanks

How many rolls 'till n values in a row.

lucasks — Wed, 19 Dec 2007 15:01:26 -0800

Given a fair die with s sides, what is the expected number of rolls r before I get o outcomes with the same value v in a row? [Note this is emphatically NOT a homework problem; I just really want to know, feel like I should be able to figure it out, can't, and it is driving me nuts.] For clarity a sample case would be, how many times do I need to roll a 6 sided die before I get 5 1s in a row. Please help.

What are the odds that I can solve this problem?

reformedjerk — Tue, 19 Jun 2007 07:46:07 -0800

Probability/stat question: I'm looking for patterns in a protein sequence, and I've found a few that occur quite frequently. How do I know these are actual patterns and not just an artifact of random amino acid distribution? I have a roughly 1000 amino acid sequence, and I've used a sliding window to chop it up into overlapping 6-mers. Some of these 6-mers occur much more frequently than others and I suspect they have some sort of biological significance. Unfortunately, I don't know to test whether these are true pattern in the biological sense, or if they could just as easily have been the result of random distribution.

I've tried comparing the expected frequency of these 6-mers based on the amino acid distribution with the observed frequency; but the chance of getting any given 6-mer randomly is so low that almost anything I observe (even the ones that only show up once) seem really significant. I'll be happy to clarify things if this post is a bit messy.

Modal verbs can versus could

stavrosthewonderchicken — Tue, 12 Dec 2006 18:20:35 -0800

English grammar: 'could be Xing' versus 'can be Xing' -- how can we explain why one is correct and one isn't? (Somebody fire up the languagehat signal!)

I'm trying to figure out a reasonably succinct structural way explain why one of these is just fine, but one isn't.

We're using modals here to express probability or guessing. Scenario: you call your wife at home, but there's no answer. You guess that she might be in the bathroom, and express it by saying "She could be taking a shower." I say this is correct.

Saying "She can be taking a shower" in this situation, I call incorrect, but I can't seem to hit on any plausible explanation for why, other than that it's just common usage. That's a good enough answer, perhaps, but I'd like to actually understand what's happening here, structurally. It could be that my brain is just wobbly today, and I'll figure it out as soon as I post this, but it's driving me a bit nuts.

Infinite vs Infinite

777 — Tue, 21 Nov 2006 13:35:13 -0800

ProbabilityFilter: Calling all mathematicians... I am not a math person, but this is something I've always wondered/had trouble wrapping my head around:

Imagine you are standing on a football field, there is a trash can about 30 yards away, and you are holding an average sized stone.

I submit that there are _infinite_ ways (different trajectories) of tossing the stone such that it lands in the trash can.

I also submit that there are _infinite_ ways of of tossing the stone such that it doesn't land in the trash can.

Surely the probability of missing is higher than making the shot? Yet if both are infinite, are they not the same probability? Are there not, in fact, infinite ways to achieve the former? Or is there something else I'm not getting?

Extra Credit: Does the math change at all if I'm _trying_ to make the shot vs. randomly throwing the stone in any direction?

What are the odds of that?

jaimev — Fri, 17 Nov 2006 07:47:20 -0800

A regular deck of 52 cards is shuffled and turned face up one by one. What are the odds of going through the whole deck and finding at least one set of two consecutive cards which have the same value? (ie, a pair). Over 30 years ago, I went through a period in 5th or 6th grade where I became fascinated by how it seemed like I could never end up running through the whole deck without encountering at least 1 pair. It was like magic to me. Someone out there must be able to easily calculate the exact probability of this and put a number to my wonderment.

Looking for help with a question of probability.

rifflesby — Fri, 01 Sep 2006 12:47:07 -0800

MathFilter: Okay, I've got this gumball machine, which contains an infinite amount of gumballs, evenly distributed between 100 different flavors... Unfortunately, the machine is broken, and three-quarters of the time I turn the crank, I get nothing. When the machine does deliver, it has a 1 in 20 chance of giving two gumballs instead of just one.

After how many cranks would I have a 50% chance of having at least one gumball of every flavor? How many for a 99% chance?

Please show your work, as I'm trying to learn how to answer questions like this for myself. :)

Philly or SF, which is safer?

rorycberger — Thu, 10 Aug 2006 11:14:30 -0800

Which city is safer: San Francisco or Philadelphia? I know there are a million "safest/least safe" cities lists available, but most of them are based only on violent crime statistics. I'm looking for a comprehensive assessment that factors in things like the chances that an earthquake is going to destroy san francisco tomorrow, the chances that terrorists drop a bomb on philadelphia, the odds that I'd get hit by bus, struck by lightening, etc. Also I'd like to factor in the proximity to other safe/unsafe cities, just in case I happen to get lost and end up in, say, Camden or Oakland. For purposes here, I'm only concerned about things that will kill me, not so much things that might wound or emotionally scar me. All things considered, where would I be safer?

[yes, this is to settle a bet]

collect 'em all - what are the odds?

ab3 — Fri, 02 Jun 2006 09:24:14 -0800

4 different models of a toy are sold in identical, sealed boxes. Assuming the 4 different models are produced/sold/distributed in equal numbers, etc..., what are the chances of buying only 4 boxes, chosen at random (at different times, even), and getting all 4 different toys? It happened to me. When I got the last one I felt like I had won the lottery. Seems like the chances would be pretty slim - I know this is probably about as basic as probability questions go, but I'm no mathematician. And if you're interested, the toys in question were Gary Baseman's collectible vinyl Fire Water Bunny series:

What are the odds that I'm such a loser?

TonyRobots — Fri, 26 May 2006 19:47:13 -0800

Probability filter: How do you determine the odds of being extremely unlucky? I know I've seen this explained, but my google-fu fails me, as does my brute-force mathematical figuring. Here's my problem -- if there's a game in which you're supposed to win 12% of the time, what are the odds that after 150 plays you only win 3 times (i.e. 2% of the time)?

Binomial distribution comparison

wanderingmind — Thu, 25 May 2006 15:54:16 -0800

Probability of one binomially-distributed variable being greater than another one - is there a closed form for this? Specifically, the problem is this. (No, this isn't for homework)

I've got two random variables, X and Y, both drawn from binomial distributions. p is the same for each, but n differs; essentially, both X and Y are the number of heads that come up in a single instance of n coin flips if the coins are weighted such that there's a p probability of heads. X and Y are independent, and n can differ from one to the other (so call the two Ns n_x and n_y).

Is there a way to get a closed form in terms of p, n_x, and n_y for Pr[X≥Y]? I've been banging my head against this for hours and I thought I'd gotten somewhere but belatedly realized I'd made a mistake. I'm hoping some MeFi combanitorics whiz can help me out here.

Math question

TheFeatheredMullet — Fri, 05 May 2006 11:49:58 -0800

Math questions about permutations for those so inclined... So, I got into a discussion with a guy here at work, and we are both stumped. We were trying to figure out how long it would take to go through all the possible combinations on one of those door locks with the 5 buttons.
At first, I figured that it would be simple, as each button is either pressed or not, making this a kind of binary problem which I thnk can be solved with 2 exponent 5, however, the order in which they are pressed matters. From what I understand, that makes this a question of Permutations, but my search for how to solve this is leading me in circles because when they talk of numbers in the explinations, they multiply each by the remaining possibilities (eg. 23*22*21) or, in the case where the numbers are replaced it is just straight multiplication (eg. 10*10*10 for three numbers between one and ten appearing in a specific order)
My case is different though, because each button can only hav 2 possible states, but if I apply the same rule, I get 2*2*2*2*2 which is the same as 2 exponent 5, however (and I could be wrong) that would be the same answer as if order did not matter.
What is wrong with my logic?

How many exam topics should be covered?

kenaman — Tue, 25 Apr 2006 04:03:57 -0800

There are 10 possible exam topics. 6 will appear on the paper. 2 questions must be attempted. How well is an ass covered based on the number of topics covered? Assume the 6 are selected from the 10 at random.
From covering 10 topics to just 2, what are the answers to the following questions?

What level of topic choice is ensured in the exam?
What levels of topic choice are likely?
What is the risk of not having covered two topics which appear?

I feel that there is a lot of extra useful data which could come out of this question. If you feel there is another angle which I have not requested please feel free to show me the data.

Will The Combined Score Of Tonight's Game Be Odd Or Even?

The Bellman — Wed, 19 Apr 2006 09:09:08 -0800

Baseball Score Probabilities: Is it any more likely that the combined score of a Major League Baseball game will be even than odd (or vice versa)? At first glance it would seem that the liklihood of the combined number of runs scored by both teams being even is equal to that of it being odd. But the game can't end in a tie, which means that some even scores cannot occur. Does this mean, as a practical matter, that an odd combined score is more likely? If the game goes into extra innings is it, as a practical matter, likely that one team will win by a single run, making an odd combined score that much more likely? Baseball and probability fans, please hope me!

How to calculate "average" probability?

futility closet — Thu, 13 Apr 2006 05:57:53 -0800

If there's a 20 percent chance of rain on Monday, a 20 percent chance on Tuesday, and a 30 percent chance on Wednesday, what's the overall chance that it'll rain during this three-day period? How do you calculate this?

NCAA's as a Probability Primer

JPD — Mon, 13 Mar 2006 18:51:41 -0800

So lets say I've got probabilities for 64 teams... So I've got the pythagorean winning % for each team in the tourney. Using one of the Bill James Sabermetric formulas "log5" I can supposedly forecast the likelihood of one team winning over another using these #'s. The formula works out to P(A) = (A-A*B)/(A+B-(2*A*B). Right all well and good. Then I think it should follow that the Probabiity of Team A winning in the second round should be - (P(C)*((A-A*C)/(A+C-(2 *A*C)))+P(D)*((A-A*D)/(A+D-(2*A*d))))*P(A). And similarly for the other teams in the second round. My problem is that as we advance to the Round of 16 there are eight different possible outcomes, Round of 8 16, etc. My question for you - how can I simplify these probabilites? I'm sorry if my question is unclear or my notation is garbage. thanks for your help

Calculating probability with children

carterk — Mon, 06 Feb 2006 21:22:32 -0800

Looking for ideas for teaching probability to kids. I've a small group of math-loving fifth graders for whom I'm trying to extend my curriculum's rather lame probability unit, which primarily deals with coin tosses and dice. I'm trying to think of some interesting things that we could reasonably calculate probabilities for.

My initial and probably best idea was to examine the probabilities that a given 5th grader might actually become a professional basketball player. It wouldn't be that hard to find the number of (American) players there are, how many other (American) kids there are who would potentially compete for those positions, and what fraction of them might be qualified and inclined to do so (maybe a poll for this last data point).

But I'm guessing there's a better idea out there, and fabulous resources that will make this easy and fun to teach. Is there? Are there?