Does "1 in a million" stuff happen?
March 15, 2011 3:00 AM Subscribe
Suppose that an event has a 1 in a million chance of happening in 1 year. The chance that it will happen once in a million years is %63. Explain.
The above (paraphrased) comes from "Why Evolution is true", by Jerry Coyne.
I've wondered about this for a while. It seems superficially intuitive that the chance should be %100. Of course this also seems completely wrong, and would fail the gambler's fallacy test at least.
Could someone explain the maths that shows the chance is %63?
The above (paraphrased) comes from "Why Evolution is true", by Jerry Coyne.
I've wondered about this for a while. It seems superficially intuitive that the chance should be %100. Of course this also seems completely wrong, and would fail the gambler's fallacy test at least.
Could someone explain the maths that shows the chance is %63?
Sorry, that doesn't answer "why 63%?", but it should show why your intuition about the 100% is wrong.
posted by EndsOfInvention at 3:22 AM on March 15, 2011
posted by EndsOfInvention at 3:22 AM on March 15, 2011
You want a binomial distribution calculator. The key is you're implying that you want to know the probability of a single success in a million trials where the probability of success for each trial is 0.000001
posted by singingfish at 3:24 AM on March 15, 2011 [1 favorite]
posted by singingfish at 3:24 AM on March 15, 2011 [1 favorite]
Yes, seems like a classic binomial problem; p = 1/1E6, n = 1E6 and k = 1. However, I get a 63% chance of the event occurring "at least once". The chances of it occurring exactly once is 34%, "once or more" is 63%.
posted by swordfishtrombones at 3:33 AM on March 15, 2011 [2 favorites]
posted by swordfishtrombones at 3:33 AM on March 15, 2011 [2 favorites]
Response by poster: swordfishtrombones: you are right, it says "at least once".
posted by antiquark at 3:39 AM on March 15, 2011
posted by antiquark at 3:39 AM on March 15, 2011
Best answer: The way to do it is to calculate the chance of it NOT happening at all. In one year, that's 0.999999. In two years, that's 0.999999^2, and so on. So, in a million years, the chance of it not happening is 0.999999^1000000 = 0.368, and the chance of it happening is 1 - 0.368 = 0.632.
posted by alexei at 3:42 AM on March 15, 2011 [16 favorites]
posted by alexei at 3:42 AM on March 15, 2011 [16 favorites]
Response by poster: EndsofInvention: I can clearly see that, hence I said "superficially intuitive". I'm fascinated at how probability is counter-intuitive. Monty Hall, etc.
posted by antiquark at 3:44 AM on March 15, 2011
posted by antiquark at 3:44 AM on March 15, 2011
singingfish and swordfishtrombones have it right. This is a binomial distribution issue. Let's try some examples, shall we?
Let's talk about flipping coins. If you flip a coin once, what's the chance that you'll get a heads? 50%. How about if you flip it twice? As you say, 100% might be the "intuitive" choice (depending on your definition of intuitive), but it's certainly wrong.
There's an easy way to calculate the actual probability though: the only way that heads wouldn't come up is if you got two tails in a row. The probability of that is easy to calculate; it's 25% (.5 * .5). The other 75% of the time, you'll get at least one heads.
Now, let's expand to dice rolls. If you roll a die once, you'll get a six 1/6th of the time. If you roll it six times, what's the probability you'll get a six? Well, the probability of not getting a six is 5/6 each time, so the probability of not getting a six six times in a row is (5/6)^6 = 33%. So you'll get at least one six the other 100-33=67% of the time.
With one million, it's the same thing. (1-1/1000000)^1000000 is the probability of not getting your event one million times. Subtract that from 100%, and you've got your answer.
On preview, or what alexei said.
posted by jweed at 3:45 AM on March 15, 2011 [4 favorites]
Let's talk about flipping coins. If you flip a coin once, what's the chance that you'll get a heads? 50%. How about if you flip it twice? As you say, 100% might be the "intuitive" choice (depending on your definition of intuitive), but it's certainly wrong.
There's an easy way to calculate the actual probability though: the only way that heads wouldn't come up is if you got two tails in a row. The probability of that is easy to calculate; it's 25% (.5 * .5). The other 75% of the time, you'll get at least one heads.
Now, let's expand to dice rolls. If you roll a die once, you'll get a six 1/6th of the time. If you roll it six times, what's the probability you'll get a six? Well, the probability of not getting a six is 5/6 each time, so the probability of not getting a six six times in a row is (5/6)^6 = 33%. So you'll get at least one six the other 100-33=67% of the time.
With one million, it's the same thing. (1-1/1000000)^1000000 is the probability of not getting your event one million times. Subtract that from 100%, and you've got your answer.
On preview, or what alexei said.
posted by jweed at 3:45 AM on March 15, 2011 [4 favorites]
While I'm here: you might be wondering, "hey, there was a 75% chance in the first example, a 67% chance in the second example, and a 63% chance in the third example. How low can this number get?" In other words how low can the probability of a one-in-an-x chance event happening at least once in x years be?
The answer is that as x goes to infinity the limit approaches 1-1/e = 63.2%. That's already pretty close to the answer for one in a million (in fact, it's pretty close to the answer for one in a hundred!), so basically if you're looking at a 1% chance over 100 years, a .1% chance over 1000 years, a one-in-a-million chance over a million years, a one-in-a-billion chance over a billion years, etc., the probability of having at least one occurrence of your event is always about 63%.
posted by jweed at 4:10 AM on March 15, 2011 [15 favorites]
The answer is that as x goes to infinity the limit approaches 1-1/e = 63.2%. That's already pretty close to the answer for one in a million (in fact, it's pretty close to the answer for one in a hundred!), so basically if you're looking at a 1% chance over 100 years, a .1% chance over 1000 years, a one-in-a-million chance over a million years, a one-in-a-billion chance over a billion years, etc., the probability of having at least one occurrence of your event is always about 63%.
posted by jweed at 4:10 AM on March 15, 2011 [15 favorites]
People have covered this question pretty well. But you might be wondering if there's some way to get the number 1 out of this question. And there is -- it's the average number of times that the one-in-a-million-years event will happen in a million years. The number of times the event occurs follows the Poisson distribution with parameter 1. (People are saying it's binomial, and strictly speaking it is, but you can approximate certain binomials with Poissons.) So its probability of occurring k times is 1/(k! e), where k! = k*(k-1)*(k-2)*...*1 is the factorial. What happens is that the cases where the event occurs more than once bring the average up.
posted by madcaptenor at 8:28 AM on March 15, 2011
posted by madcaptenor at 8:28 AM on March 15, 2011
Sorry -- I didn't close the link in my previous comment.
posted by madcaptenor at 8:29 AM on March 15, 2011
posted by madcaptenor at 8:29 AM on March 15, 2011
This thread is closed to new comments.
If you roll a 6-sided-die six times, are you 100% guaranteed to roll a 6 at least once?
If you have an event that has 1 in a million chance of happening, and the event has a chance to occur 1,000,000 times in a row, are you guaranteed for the event to happen at least once?
posted by EndsOfInvention at 3:21 AM on March 15, 2011 [1 favorite]