William M. Briggs

Statistician to the Stars!

What Are The Chances Of That?

A random Wikipedia picture about dice. Because posts have to have pictures.

Go ahead and ask me. I’ll even pretend to laugh. Just as I chuckle warmly with you when you quote Mark Twain (or Disraeli) abut damned lies.

(Secretly, I’ll be looking for the nearest exit. But, oh yes, I’ll laugh.)

Whenever any weird, comical, or seemingly unusual event happens—like when I was walking with a friend and a school bus drove by and a kid threw a crayon out the window which landed plop into his pop1—and a guy like me is around, somebody hilariously asks, “What are the chances of that!”

Here’s the universal answer: if it happened, then the chance is 100%. I’m not kidding. That’s the real answer. Why? Because all chance is calculated conditional on circumstance. And the circumstances were whatever it was that caused the strange thing to happen. And since the strange thing happened (as it was caused to) the chance is 100%.

If that sounds like a lawyer’s answer that’s because nobody ever means the question in this sense. They want to know what are the chances the strange thing happens again.

Same answer: 100%. It’s the same answer for the same reason. Because all chance is calculated conditional on circumstance, and the question does not specify any change in circumstance, the probability must also remain unperturbed.

Probability isn’t alive, it is a zombie, a perfect slave to the information which specifies it. If nothing changed, if the world really were to reset itself to the precise same condition before the strange thing happened, and there is no change to the physical laws and (all) causes which drive everything, then the strange thing must happen again.

But nobody means the question in this sense either.

So what sense do they mean it? That’s not clear. Two things are needed in any chance or probability calculation. The first is the proposition of interest, which is our strange thing. When somebody asks “What are the chances?” they clearly mean to keep this proposition intact. The second thing, as discussed, are the conditions. We’ve already seen that we’re not meant to keep these conditions precisely identical, so people must be imagining these undergo some sort of shift. What kind?

Let’s switch examples in the hopes of clarification: an American roulette wheel. The proposition is “The ball lands on the 5″, which isn’t especially strange, but the proposition isn’t the problem. What are the circumstances? A bunch of slots (38) on a wheel, one of which labeled “5″, and a ball which must land in one of the slots. Probably, based on this and no other evidence (“no other” means no other), is 1 in 38.

Now if the ball in fact landed on 5, and somebody asks, “What are the chances of that happening again?” we’d say “One in thirty-eight”. Why? Not because we expect the circumstances to be identical on the next spin—because if they were then a 5 must again show—but because we keep the same evidence in mind (that the wheel has 38 slots, etc.).

Inveterate gamblers start the same but change the evidence. They say to themselves something like, “It just came up 5, it probably won’t do it again. Of course, the wheel still has 38 slots and so forth.” Now the probably the inveterate gambler then calculates based on this changed evidence probably won’t be zero, but he’d say it was close.

And that’s okay, because he did change the evidence (as I cannot emphasize enough). A pedant would wag his finger at the gambler and insist he’s using the wrong evidence. Well, and so he is, but the gambler’s not coming to the wrong probability based on the evidence he used.

The best evidence to use nobody has: that would be the full physics of the circumstance (weight of wheel, forces on the ball, friction of the slots as ball hits, and on and on). Now this sounds a bit like listing the circumstance for the strange thing, doesn’t it? But if we had that information, we’d know whether the 5 was going to show with certainty.

So where do we stand? For the strange thing, the asker means to change the circumstance/evidence, as we saw, but he never specifies how to change it. We know the circumstances must be different to get a probability different than 1, we just don’t know in what direction they should be modified.

Since we can’t know what aspects of the circumstances should change, there’s no way to know what the probability of the strange thing is. Consider the crayon. Should the bus be moving or stationary? At what distance is the cup? What color crayon? Should color matter? Which angle is the cup mouth facing the bus? Is the cup moving or stationary? And on and on endlessly. Unless we are prepared to say exactly which evidence we’re considering, there’s no way to know the chance.

What a disappointing answer.

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1That’s right. I said pop. You barbarian.

19 Comments

  1. “So what sense do they mean it?”

    Your whole post goes off the rail when you try to answer the statement as if it were a question.

    Let’s look at the statement again.

    “What are the chances of that!”

    Notice that it ends in a exclamation point not a question mark. It isn’t meant as a question at all, it is a declarative statement.

    If you have trouble understanding that statement as a declarative statement rather then a question, here is a rough translation: “That was weird!”

  2. Briggs

    4 March 2014 at 9:57 am

    MattS,

    You’re right. Nobody ever wanted to know the chance. I’ll delete the article when I have some free time.

  3. a school bus drove buy

    Someone purchased a hamburger at a drive-thru while riding in a school bus?

    We’ve already seen that we’re not meant to keep these conditions precisely identical, so people must be imagining these undergo some sort of shift. What kind?

    The conditions are mostly unknown. Pretty hard to specify how an unknown might change. I think people really mean “What are the chances it could have happened?” even when they are saying did happen.

  4. Briggs

    4 March 2014 at 10:42 am

    DAV,

    My enemies are out in force!

    Yes, people might mean “could have happened”, but this is equivalent “it did happen”, because “could have” implies the very same change in circumstance, i.e. evidence, the specifics of which are left unstated.

  5. Briggs,

    Perhaps but “could” rather implies without knowing the outcome. Even after the fact, people have the innate feeling that the confluence of circumstances were leading to the result likely will not happen again but are otherwise unpredictable. Most people, that is, there always seems to be someone who will bet on the last roulette outcome apparently assuming the circumstances will NOT change.

  6. If I buy a single lottery ticket, what is the probability that I will win? One in a bazillion or something.

    If I buy a single lottery ticket, what is the probability that somebody will win. Maybe one in 10 or one in 20 because lots of people buy lottery tickets.

  7. Oh my! circumstances were leading to the result
    Seems your enemies don’t discriminate.

  8. I like the post for the same reason I find the blog so interesting. I confess to having read too much Wittgenstein while in the nursery but careful attention to propositions, definitions and clarity about what we know or think we know is always welcome.

    An exclamation (!) can be both an exclamation and a question or vice-versa depending on the context and the tone of voice. “We won the game,” could be a simple statement, a question of surprise or an exclamation of astonishment or understood by the listener as a question and an exclamation. Other examples could be added involving expletives but consideration of them will be left to the reader’s imagination.

  9. This post got me thinking about my own probability situation. I’ll start with a common one where the human mind has trouble. If you’re in a room, what’s the minimum number of people required where there is a better than 50% two people have the same birthday? 23! That’s a lot lower than most people would suspect.

    As well, when we get to 100 people, the chances of two having the same birthday are 99.99997%. Most people start this problem with the basic chance of any two people having the same birthday as 1/365. It’s just hard to intuit non-linear relationships.

    As a side note, my wife and I have the same birthday in the same year. I’ve been trying to figure out those odds but I haven’t come up with a good method. Any help?

  10. mpcraig,

    “As a side note, my wife and I have the same birthday in the same year. I’ve been trying to figure out those odds but I haven’t come up with a good method.”

    100%!

    But your question really is, what is the chance that any married couple shares the exact same birthday?

    I suppose we could look at census records and get the fraction of married couples have the same birthday.

    or you can build a model.

    The key assumption then is, what is range of acceptable age differentials? This is a personal question, and probably depends upon your age. So, we make take a guess and say +2 years to – 8 years giving you a 10 year range.
    We can do an accounting for leap years, but I don’t think it is worth the effort with given the margin of error above. Instead round it to something convenient…

    ~ 1/3500

  11. mpcraig — people tend to hang out with and marry other people who are about their same age for lots of reasons — attended school together for example.

    The first thing you have to do is state the problem clearly. The probability of you and your wife having the same birth year is 1. I don’t think that is the question you want answered.

  12. It is 100% certain that somebody will win the lottery. After the fact you can give the likelihood by counting how many tickets were sold. And the odds that i did not win the jackpot.

  13. Something similar is some who argue in favor of freewill will often make the case that a certain decision could have been made otherwise. This is in error. What you decide to do at a given moment can never be changed or revisited – and no matter how similar a future scenario might be, it will not be identical.

  14. “It is 100% certain that somebody will win the lottery.” Not true. Some weeks nobody wins and the pot rolls over to the following week. Sometimes there’s a double rollover. That is, with respect to the jackpot which you mentioned.

  15. @DougM, in response to @mpcraig: Good approximate analysis of the intended question – which should be “What is the probability that the random selection of a married couple in North America would produce two people who were born on the same day?”

    The commonly used non-technical wording “what is the chance that any married couple shares the exact same birthday?” is ambiguous as it could refer either to a particular couple – in which case, as Briggs will happily tell you, the answer is either 100% or 0 depending on the actual facts for that couple, or it could mean what is the chance that there exists any couple with shared day of birth – in which case the existence of mpcraig proves it is 100%, or it could be interpreted in the sense where “for any” is used as a substitute for “for all” – in which case most of mpcraig’s friends provide counterexamples and so the probability of it’s being true is 0.

    What I am waiting for is Briggs to provide an example of a statement in which “probability” or “the chances” is well-defined in some practical context by something other than some kind of frequentist approach.

  16. In the spirit of Fermi and DougM, one might answer Briggs’ friend’s question as follows:

    The friend was presumably not surprised to find himself walking with a cup of pop nor to see garbage get thrown from a schoolbus, so what he probably meant (or at least what *I* would mean in the unlikely event that Briggs would consent to be seen walking in the company of a scruffy old atheistic socialist frequentist like me) is “What is the probability of a randomly thrown piece of garbage ending up in a cup that happens to be nearby when it is thrown?”

    Following the approach of DougM we might estimate the distribution of landing sites for an item thrown from a vehicle as uniform in a semicircle of radius 2 meters and the opening area of the cup as a circle of radius 4cm and taking the area ratio gives a probability of about .0008 or a bit less than one in a thousand. So we can expect to have to wait on average for a thousand or so pieces of garbage to get thrown near us before actually catching one in our cup.

    Of course your risk of actually catching garbage in your cup on a walk with Briggs depends on the kind of place he takes you. If you want to minimize the risk of falling garbage you can come with me for a nice walk in a park by the sea – but given the habits of our seagulls I can’t promise that you won’t stand some risk of doubling the ‘o’ in your pop.

  17. Rich, well if you include the unsold tickets in the raffle, then indeed there is a fat chance that nobody wins. It just lowers your odds substantially.

  18. Or does it increase your odds (?)

  19. Charles Boncelet

    5 March 2014 at 11:06 pm

    I’m not sure I can disagree with this post any more than I do.

    When someone asks “What are the chances…” they don’t want an answer of 100%. Clearly that’s not what they mean. What they are asking is “what is the chance those unusual circumstances will arise again?” given the usual evolution of things. That unusual set of circumstances may never occur again, or it may a limited number of times, or maybe an infinite number of times. The problem is to determine which of these are true.

    Coincidences happen more often than expected because the number of possible coincidences is astronomical.

    Re: the roulette wheel. If all you know is the wheel has 38 slots and one slot comes up, and that’s all you know, then you don’t know enough to quantify the probabilities. They could be anything consistent with the two basic rules (probabilities are nonnegative and sum to one).

    If you have additional information that the slots are approximately equal sized and the wheel is spun in such a fashion as to not prefer any slot, then equally likely is a reasonable starting point. (I’d start there and adjust my probability estimates as I see more data. After all, probabilities are only meaningful as to their ability to predict the future.)

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