Comments on: What are the odds of finding 20 needles in 270 haystacks?
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks/
Comments on Ask MetaFilter post What are the odds of finding 20 needles in 270 haystacks?Wed, 01 Dec 2004 07:44:08 -0800Wed, 01 Dec 2004 07:44:08 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: What are the odds of finding 20 needles in 270 haystacks?
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks
Suppose there are 20 objects hidden in a collection of 270 haystacks. What are your odds of searching 100 haystacks (assuming that if you choose to search a haystack with an object in it, you'll find that object) without finding an object? My discrete math skillz have eroded to the point where any attempt to work this out myself yields preposterous numbers..post:ask.metafilter.com,2004:site.12359Wed, 01 Dec 2004 07:40:26 -0800COBRA!mathmathematicshaystacksprobabilityBy: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215052
I think it would be (80!-60!)/(270!-250!)<br>
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Do I pass the test?comment:ask.metafilter.com,2004:site.12359-215052Wed, 01 Dec 2004 07:44:08 -0800DoohickieBy: GrumpyMonkey
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215053
As a follow up to this - what are the chances that you would not find one of those 20 objects after unrolling 100 haystacks over an 8 hour period?comment:ask.metafilter.com,2004:site.12359-215053Wed, 01 Dec 2004 07:44:25 -0800GrumpyMonkeyBy: andrew cooke
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215058
can you have more than one object in a single haystack?comment:ask.metafilter.com,2004:site.12359-215058Wed, 01 Dec 2004 07:47:59 -0800andrew cookeBy: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215063
No, those minus signs should be division signs.<br>
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If I do my math right, that gives you 4x10-12 (4 times 10 to the minus 12th power).comment:ask.metafilter.com,2004:site.12359-215063Wed, 01 Dec 2004 07:49:36 -0800DoohickieBy: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215064
Naw, I'm still screwed up.comment:ask.metafilter.com,2004:site.12359-215064Wed, 01 Dec 2004 07:50:58 -0800DoohickieBy: naxosaxur
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215065
COBRA!, i know what you did <a href="http://www.tvtome.com/tvtome/servlet/GuidePageServlet/showid-6686/epid-379204/">last night </a> last night.comment:ask.metafilter.com,2004:site.12359-215065Wed, 01 Dec 2004 07:51:44 -0800naxosaxurBy: smackfu
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215067
At the end there were only 12 objects to find, since 8 of the objects were removed by others and not replaced. That hurts the odds and needs to be reflected.comment:ask.metafilter.com,2004:site.12359-215067Wed, 01 Dec 2004 07:52:49 -0800smackfuBy: dual_action
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215068
God, I felt bad for her. I would have given up about hour two.<br>
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I can only think there must have been one, she just missed it. I can't imagine how hard it would be to search in the dark.comment:ask.metafilter.com,2004:site.12359-215068Wed, 01 Dec 2004 07:53:13 -0800dual_actionBy: vraxoin
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215069
Listen, dude. Stop fucking around with probabilites and just get out there with a pitchfork. You'll find those objects.comment:ask.metafilter.com,2004:site.12359-215069Wed, 01 Dec 2004 07:53:45 -0800vraxoinBy: briank
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215071
I was dumbstruck that they could not find a clue packet after eight hours of unrolling the bales.<br>
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(For those who don't watch, the question refers to a challenge in this week's episode of "The Amazing Race")comment:ask.metafilter.com,2004:site.12359-215071Wed, 01 Dec 2004 07:54:29 -0800briankBy: golo
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215073
If there can be one and only one object per haystack, then the chances of finding an object in any random stack would be 20/270 (0.074). The chances of picking a random stack and NOT finding an object then would be 1-0.074= 0.925 (good chances). The chances of this happening twice would be 0.925 squared (0.857), for three stacks would be 0.925 cubed and so on. If I'm right the chances of choosing 100 haystacks and not finding simngle object would be (1-(20/270))^100=4.54e-4, or something like 0.00045. Not very likelycomment:ask.metafilter.com,2004:site.12359-215073Wed, 01 Dec 2004 07:57:29 -0800goloBy: pardonyou?
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215074
I found it distinctly unfair to end an elimination leg with a pure luck challenge. They've done that before, and it just infuriates me. <br>
<br>
(still one of the best shows, though)comment:ask.metafilter.com,2004:site.12359-215074Wed, 01 Dec 2004 07:57:43 -0800pardonyou?By: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215075
250/270 x 249/269 x 248/268 x ... x 151/171 = 3.8 x 10-9(3.8 times 10 to the minus 9th power)comment:ask.metafilter.com,2004:site.12359-215075Wed, 01 Dec 2004 07:58:15 -0800DoohickieBy: COBRA!
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215078
smackfu's right, but I wanted to leave out that complication until the easier version was worked out.<br>
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And only one object per haystack, andrew cooke.<br>
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<small> It's important to me to work this out, because my wife had ten bucks riding on Lena and Kristy, and I'm trying to console her with the knowledge that she's the victim of a statistical freak accident. </small>comment:ask.metafilter.com,2004:site.12359-215078Wed, 01 Dec 2004 07:59:23 -0800COBRA!By: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215079
golo: With each haystack checked, the denominator decrements by one. For each pick, there are (n-20)/n possible good picks, and n goes from 270 to 171 (one hundred trials)comment:ask.metafilter.com,2004:site.12359-215079Wed, 01 Dec 2004 08:00:59 -0800DoohickieBy: COBRA!
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215080
<em>250/270 x 249/269 x 248/268 x ... x 151/171 = 3.8 x 10-9(3.8 times 10 to the minus 9th power)</em><br>
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That's the construction I was using, but that can't be right, can it? That seems outlandish. I guess it <em>was </em>outlandish, though.comment:ask.metafilter.com,2004:site.12359-215080Wed, 01 Dec 2004 08:02:20 -0800COBRA!By: andrew cooke
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215081
if it's a single object per haystack, then you have 250 haystacks without objects, at first. so the probability of picking an "empty" haystack is 250/270. the probability of picking another is then 249/269. and so on. so the final total probability (for 100 haystacks) is (250x249x...151)/(270x269x...171) which is (250!/150!)*(170!/270!). which is about 1 in 16000. so you are very likely to find an object.<br>
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and it doesn't matter how many hours it takes.comment:ask.metafilter.com,2004:site.12359-215081Wed, 01 Dec 2004 08:02:35 -0800andrew cookeBy: Doohickie
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215084
Yep, I agree with andrew cooke.... 1 chance in 16,223. I ran the numbers wrong the last time.<br>
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I think we have an answer.comment:ask.metafilter.com,2004:site.12359-215084Wed, 01 Dec 2004 08:06:37 -0800DoohickieBy: gyc
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215087
pardonyou?: I agree with you completely, but plenty of other teams have been eliminated due to pure luck (such as slow cab drivers and airplane delays). OK, back to the math.comment:ask.metafilter.com,2004:site.12359-215087Wed, 01 Dec 2004 08:07:17 -0800gycBy: golo
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215097
Thanks Doohickie. Apparently I was thinking of choosing 100 stacks at a time, not choosing one by one and checking their contents before continuing.comment:ask.metafilter.com,2004:site.12359-215097Wed, 01 Dec 2004 08:20:22 -0800goloBy: box
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215100
In terms of probability, it's my understanding that it doesn't matter whether you choose 100 stacks at a time or if you choose one and check before continuing. (Well, assuming you don't accidentally pick a stack you already chose.)comment:ask.metafilter.com,2004:site.12359-215100Wed, 01 Dec 2004 08:25:19 -0800boxBy: Divine_Wino
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215104
However, it's still a real world problem and I think she might have missed one or two of the clues in bales she unrolled (at one point her sister told her to look around in the already unrolled stacks which she did listlessly) what with the fatigue and the panic and the running around, so that fuzzies the odds. (as Dual_Action points out)<br>
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As long as we're on the topic that screamy dude with the died blue hair at the back of his head needs to be fired out of a cannon into a pit of rats with herpes. I want to find his wife and pay for her divorce lawyer even though I'm poor.comment:ask.metafilter.com,2004:site.12359-215104Wed, 01 Dec 2004 08:28:09 -0800Divine_WinoBy: Galvatron
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215112
andrew cooke's got it. Another way of arriving at the solution: there are C(250, 100) unique selections of 100 haystacks that have no hidden item (where C(., .) is a binomial coefficient). There are C(270, 100) total unique selections of 100 haystacks. So the probability of not finding an item is C(250, 100) / C(270, 100) = (250!/100!/150!) / (270!/100!/170!) = (250!/150!)*(170!/270!)comment:ask.metafilter.com,2004:site.12359-215112Wed, 01 Dec 2004 08:35:47 -0800GalvatronBy: saintsguy
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215160
There is an additional assumption being made that if you search a haystack, then you are guaranteed to find the object hidden within it.<br>
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Of course, this is completely untrue, hence "needle in a haystack".<br>
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This complication shortens the odds of not finding a single clue.comment:ask.metafilter.com,2004:site.12359-215160Wed, 01 Dec 2004 09:17:52 -0800saintsguyBy: yerfatma
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215162
This was on in the background last night while we were applying for mortgage refinancing. My contribution to the financial pursuit went something like this:<br>
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"She's still fucking looking . . . still looking . . . oh my God, the old people caught up to them . . . it's dark now."comment:ask.metafilter.com,2004:site.12359-215162Wed, 01 Dec 2004 09:17:57 -0800yerfatmaBy: Mayor Curley
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215178
<i>As long as we're on the topic that screamy dude with the died blue hair at the back of his head needs to be fired out of a cannon into a pit of rats with herpes.</i><br>
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Send the guy with the two little bundles of hair on his forehead with them. And I'm very worried that the old couple and the father/daughter team will get eliminated and there will be no one to root for.comment:ask.metafilter.com,2004:site.12359-215178Wed, 01 Dec 2004 09:37:35 -0800Mayor CurleyBy: COBRA!
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215198
You can always watch and root for Bolo to spontaneously combust. It's going to happen.comment:ask.metafilter.com,2004:site.12359-215198Wed, 01 Dec 2004 09:58:38 -0800COBRA!By: drewbeck
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215262
GRANDPARENTS!!!comment:ask.metafilter.com,2004:site.12359-215262Wed, 01 Dec 2004 11:16:54 -0800drewbeckBy: spilon
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#215413
I dunno the answer to the math problem, but who knew wathching pro wrestlers trying to count stuffed animals and non-stick bakeware could be so damn entertaining!comment:ask.metafilter.com,2004:site.12359-215413Wed, 01 Dec 2004 14:42:42 -0800spilonBy: amarynth
http://ask.metafilter.com/12359/What-are-the-odds-of-finding-20-needles-in-270-haystacks#219793
Here's a <a href="http://www.tarflies.com/article.php?_f=detail&id=397&s_id=6&ep_id=172&tarfly=863aaa54221b9f38c1229bcf62665ced">game</a> that let's you relive Lena's search from the comfort of your own home.comment:ask.metafilter.com,2004:site.12359-219793Tue, 07 Dec 2004 12:57:50 -0800amarynth