Many are familiar with the birthday probability problem, but it might be new to some of you. Given a group of people chosen at random, what is the probability that two or more of them celebrate their birthday on the same day of the year? To simplify the problem, assume 365 day years and each day as likely as any other day (these assumptions are only approximately correct). The following are some sample probabilities. When I first encountered this one, it seemed counterintuitive to me. P(22) = .475 695 (Less than 50-50) P(23) = .507 297 (Better than even money) P(30) = .706 316 (Odds 7 to 3 in favor) P(40) = .891 232 (Almost 9 to 1 in favor of a hit) If you flip a true coin 4 times, the probability of THHH is the same as the probability of HHHH, both probabilities are equal to 1/16 or 15 to one against a hit. If you flip a coin many times and record the sequence of heads and tails, what is the probability of THHH occurring before HHHH? Answer 15/16 or 15 to one in favor of THHH occurring first. If you flip a coin millions of times and record the number of times that THHH occurs and the number of times that HHHH occurs, which occurs more frequently? HHHH occurs more frequently. The more frequent event is less likely to occur first. Suppose you make a set of three dice (Red, Blue, & Yellow), using the numbers from 1 to 9. Each die has three of the nine numbers, with opposite sides having the same number. The Red die has numbers 1, 6, 8; The Blue die has 3, 5, 7; The Yellow die has 2, 4, 9. You pick one of the three dice and then I pick one of the remaining two. We play a game by rolling the pair of dice chosen. If my die has a higher number than yours, I win a dollar, and vice versa. If you pick first, I will always win money in the long run. I expect to win 5 times for every 4 times you win. The Red die beats the Blue die with 5 to 4 odds or a 5/9 probability. The Blue die beats the Yellow die with 5 to 4 odds or a 5/9 probability. The Yellow die beats the Red die with 5 to 4 odds or a 5/9 probability. Is likely to win is not transitive like greater than. Gloria and I just returned from a delightful dinner which included 1.75 liters of wine and several cups of coffee heavily laced with Irish Cream Liquor (highly recommended). If I were more sober, I think I could remember some other probability anomalies. There are many of them, because probability is often counterintuitive.
May be all you are proving is that BY nature absolute chance does not exist. And that probabillity is it's pure form is only a mathematical construct not supported by evidence. Actually this is an interesting point I think. Can any one prove that chance or probablity ( as absolute) are in fact true? If Chance is disproven then what does that tell us?
Quantum Quack: While often counterintuitive, Probability Theory is as well established as Calculus, Plane Geometry, Topology, Number Theory, Theory of Equations, and many other branches of mathematics. You might claim that any or every mathematical discipline is abstract and not compatible with reality. However, all the experimental evidence indicates that the above disciplines are incredibly accurate models of reality. Otherwise, much of our technology would not function as it does. Furthermore, you can experimentally verify the above probability calculations. The coin flip probabilities are easy to simulate. Creating the dice might take a bit of work, but would not be difficult to do. The birthday calculation can easily be verified. Take a Year Book or a biographical dictionary, pick groups of people like the first 30, or the last 40, or 25 people starting with some particular person, or every third person starting at some particular person. BTW: Casinos in Las Vegas, Reno, Evian, Monte Carlo, Atlantic City, et cetera provide a lot of experimental evidence supporting Probability Theory.
I think I am not so much disputing probability theory, but suggesting that for it to be absolutely true it would have to be proven as such. And to do so would require evidence frorm reality and not just in maths. Using your reference to casino's How many casinos go bankrupt for the huge win of a gambler not anticipated for? For example. If probablitity was absolute, casino's would be absolutely perfect as a way of making money I would think. If probablity isn't absolute ( in reality ) then what does this say? To me this would suggest that the uncertainty principle of the observer influencing the outcome becomes more applicable. If this is so then the question is how much influence does the observer have on the outcome and in what way? Therefore this explains why probability may be seen as counter intuitive.
Further, It's sort of funny in a way, when you place a bet on a lottery with a chance of one in a million and yo happen to win, the probability is one in a million. But in reality is it not just 50/50 in that you either win or you loose. The cances of pulling an ace of spades from a full deck of cards is 1 in say 52 face down......with the drawn card face down in front of you. you may ask what is the probability of it being the card I want? Could the answer be 50/50 or 1 in 2?
Quantum Quack: I think you should learn about Probability Theory before making comments on the subject.
No. You seem to be assuming that if there are two options, either option will always have an equal chance or occurring. This obviously isn't the case.
Fair point and maybe by posting here I am trying to cheat a little and learn here instead. It is obvious that I have no formal learning on the subject except for basic accounting statistics and simple probablitity. But when I did study the issue the subject of counter intuitive probability stuck in my mind. For so often life seems to defy what would be considered to be most probable. And intuitivley, probability studies and reality seem to conflict. Afterall why use the terminology of "counter intuitive" in the first place? What does "counter intuitive" mean?
For so often life seems to defy what would be considered to be most probable. But considered by whom? Yourself? And intuitivley, probability studies and reality seem to conflict. This seems like the point of the thread to me. This is why our intuituion is not always a fair indicator of what one should expect. Afterall why use the terminology of "counter intuitive" in the first place? What does "counter intuitive" mean? Counter-intuitive means that which goes against our intuition. What you may expect regarding the probabilities of certain events is actually not what those probabilities really are. And if you wish to argue that probability theory conflicts with nature, you must demonstrate actual conflict and not appeal to your own intuition.
thanks, I have taken the term some what reversed. you are saying that when you look at the data the probablity analysis sometimes appears contrary to what you would expect. But only when you look at the stats. I was extending it to reality instead...My apologies. Please Register or Log in to view the hidden image!
I don't see how you figured that THHH is more likely to occur before HHHH. Can you expand on this? I also have an issue with the second bullet, but it depends on your interpretation of the first problem, so I'll wait for your response.
Re: Re: Counterintuitive probabilities. Maybe it has something to do with the fact that the start of a successful THHH sequence necessitates the failure of a HHHH sequence, so THHH is likely to show up fist; but a successful HHHH sequence can overlap with a successful THHH sequence, while a successful THHH sequence can't overlap with HHHH. To put it another way, any time you get a successful THHH sequence you are already most of the way to a successful HHHH sequence, since it will only take one more H to complete the HHHH. This means that any time you complete THHH you will have a 50/50 chance of also completing HHHH, since the next letter will be either T or H. On the other hand, there is no way to tack on an extra T to HHHH to generate a THHH sequence. Hmm…did that make any sense?
Re: Re: Re: Counterintuitive probabilities. Yes, I do follow what you mean, but I don't think that's valid. (I think you're reading the sequence in the opposite direction of Dinosaur, by the way) Saying that a THHH sequence occurs because an HHHH sequence fails is fine, but that doesn't make it any more likely to occur first. If I have HHH and I'm getting ready for the fourth flip, it's equally likely that I'll get either T or H. I do agree that the HHHH sequence can overlap THHH if you interpret an event as an individual coin flip (as opposed to 4 coin flips). This leads to a murky area that I don't think I'm qualified to enter - is it valid to use a continuous stream of single flips, using the last four as your current 'sequence'? To me, it would seem that the probabilities would be different in this case, and that this would lead to misinterpretation. But maybe I'm way off.
Re: Re: Re: Re: Counterintuitive probabilities. Look at it this way: if a T comes up, then it is impossible to complete a HHHH sequence without first completing a THHH sequence. This is why THHH is likely to come up first. However, there is a 50% chance that any THHH sequence will be followed by another H, creating a HHHH sequence. The THHH sequence doesn't have this ablility to tack its self onto a finished HHHH sequence, so although it is more likely to show up first, in the end there will be fewer of them. I think we have to assume that the blocks of 4 flips are able to overlap. If we only considered discrete set of four flips, we would have to say that THHH is just as likely to occur as HHHH, or any other sequence, and pack up and go home.
Re: Re: Re: Re: Re: Counterintuitive probabilities. Gotcha. I'd still like to see where the number came from, though.
Re: Re: Re: Re: Re: Re: Counterintuitive probabilities. There is a 1/16th chance of getting HHHH before THHH because, unless you get HHHH right off the bat, it will be impossible to complete a HHHH sequence without first finishing a THHH sequence. As soon as the first T comes up, HHHH is doomed as far as going first is concerned. 1/16 is the probability of getting HHHH as the first four numbers in your sequence.
Nasor: Your last post explained it perfectly. If the first four flips are not HHHH, THHH will occur first. Note that HHHH occurrences can be clustered. If HHHHH occurs, there are two HHHH series. Similarly for HHHHHH, which has 4 HHHH series. This cannot happen with THHH. These and other probability anomalies were discussed in a Martin Gardiner column in Scientific American many years ago. If you get a chance to obtain a book with a collection of Martin Gardiner columns, it is worth buying. His columns discussed and described all sorts of fascinating concepts. BTW: The set of three dice is based on a 3X3 magic square.
