I know that mathematical types will be wringing their hands at this - but it’s something that I don’t understand…that said…
Let’s say I flip a fair coin 10 times…and every time it comes up tails. From what I understand about standard deviations, there is a 99% chance at the end of 100 flips, that the heads to tails ratio will be somewhere between 48/52 and 52/48. Now, given that, can I not make a statement that the coin is “due” to flip heads more than tails over the next 90 flips? I know that each flip is independent, but is there really no way to combine the short term with the long term??
Long answer: Probability only discusses action that will happen in the future. The “probability” of an action happening in the past is exactly 100% or 0%: It either happened or it didn’t happen. You can’t use actions in the past to exclude possible actions in the future.
Another way of looking at it is that the probability of receiving 99 tails + 1 head is equal to the probability of receiving 99 tails + 1 tail.
Yet another way of looking at it is that, in practice, if you flip a coin 99 times and it comes up tails, it’s almost certainly not a fair coin.
Time flies like an arrow. Fruit flies like a banana.
Which gets us into the realm of statistics, where you can ask with what degree of certainty you can decide that the coin isn’t fair after this has happened. Tricky part is that you should decide what kind of trial you are going to conduct beforehand.
(Unless you’re a Bayesian, in which case you’re allowed to show up with your prejudices and let the data modify them …)
In the probabilistic model, each flip is indeed independent of the others. Thus, as noted already, just because you’ve had a string of heads is no reason to expect a higher number of tails in the next flips.
In real life, if you get a string of nine heads in a row, there’s a strong likelihood that the next flip will also be heads – either because the coin is not a fair coin, or because your flipping technique doesn’t get any spin on the coin, or some similar non-probabilistic cause.
I understand what everyone is saying - and I believe that each flip is independent. But it really seems strange that there is no way to look at the past, know what the future will be (at least to 99%), and synthesize these two…
Here’s a bad example:
If I watch a car go by a certain point on a 1 mile long circular track at 20 mph, and then see it whip past me in less than a minute, i know two things:
a) the car was moving faster than 20 mph when it past me the second time -or- was moving faster during the lap
b) the car was moving at 20 mph when it past me the first time
Therefore, I know that it sped up during that lap.
Why doesn’t this prinicple translate to coin flips. Again, I know it doesn’t, I just don’t understand why.
There is a human need to want things to work out as they should. You’re going to flip a coin 100 times. You rightfully think it should come out in the neighborhood of 40/60 through 60/40. You start flipping and you get 10 heads off the bat. You want things to even out–to reach the point they should. But no. Once those heads started coming up, you have to revise your expectation regarding the distribution of heads and tails over the 100 flips. This time that initial estimate is probably going to be off. It’s tough, but you have to let it go. Things don’t always work out as they should
Tony
Two things fill my mind with ever-increasing wonder and awe: the starry skies above me and the moral law within me. – Kant
Here is another way of looking at the problem. Let’s say I’m looking at a two sided piece of paper on which are printed the results of 200 coin flips. These coin flips have already taken place, the first hundred shown on side one and the second hundred on side two. I am told an unbiased coin was used to generate the results. I look at the first page and see 99 heads and one tail.
Do I have any reason to think that there will be more tails than heads shown when I turn the paper over and look at the other side?
Assuming that we actually do have a fair coin, the answer would be a). However, the “justification” for b) is right, too, but the conclusion is false. The long term ratio will get closer to 50/50, but that doesn’t mean it has to “bounce back” immediately. For example, if we start out with 100 straight heads, and then the next 10,000 flips are pretty much 50/50, the overall average will be close to 50/50–the first 100 will have less and less affect on the long term ratio as we continue flipping the coin.
…ebius sig. This is a moebius sig. This is a mo…
(sig line courtesy of WallyM7)
Thank you Sethdallob for asking a naive question. You have inspired a bunch of us to try to illuminate the situation.
As has already been said, certain probabilities and standard deviation existed before you began your 100 flips. After you have completed 10 of them, the conditional probabilities on the 100 flips have changed, since they include the actual results to date.
This principle is illustrated by the not very funny joke about the airplane passenger who worried about a bomb. So he carried his own bomb on board, since the chance of TWO bombs on a plane would be extrememly unlikely.
If the coin really truly is fair, and the flips independent, a) is the answer, and there is in fact an overwhelming probability that the sum of the two sides of the paper will show a much greater number of heads, reflecting the given knowledge that the first 100 flips have already come up 99 - 1 heads over tails. In fact, the result for the sum of the two sides will be distributed around a most probable result of 149 heads, 51 tails.
If you then have ten more pages, the sum will be distributed around a most probable result of 649 heads, 551 tails, and will probably still show a greater number of heads. If you have a million more pages, there’s only a slight probability that the total will have more heads than tails, reflecting the initial unusual run.
Part of the problem here is that people have heard of some fuzzy “law of averages”, and expect an abberant run to be actively “corrected” somehow. In fact, the abberant run will eventually be swamped by a large number of trials.
Note that I STARTED by saying IF the coin is fair and the flips independent. In reality, I would take the first page of results to be a damn good indicator that it wasn’t, but we’ve been over that ground already. Again, how confident I should be of that indicator, is entering into the study of statistics, rather than probability.
Actually, this isn’t an inane question - it illustrates common misconceptions about probability much better than the trickier “Monty Hall” type puzzles you see in forums like this, which also often hinge on how to properly model given knowledge.
The answer to the original question is that there are really two questions. Assuming that we’re playing with a fair coin, the odds of the next flip being heads or tail are 1/2. Most of us agree on this.
The other part, expecting a result because the results so far are above or below the expected value, is a common mistake. What’s really happening is that sethdallob is taking a sum of random variables (the results of each flip). The expected value of this sum for the coin toss example is indeed 1/2, but the variance of the sum increases with the number of tosses. In effect, this means that the average value of the results from many games can be predicted, but the error in predicting the result of any one game is likely to be huge.
This is a classic random walk problem. You would like to think that if you took random steps left and right (say, left when you get heads, and right when you get tails), that you’d stay somewhere near your starting point. Realistically, though, if you were to look at the range of steps away from your starting point that you’d be likely to end up at with 90% probability, you’d find that it increases as sqrt(N). The upshot of this is that if you let N go to infinity, the odds of landing back at your starting point go to zero.
So, back to the original question. Although the expected number of heads and tails is the same, the probability of getting the same number of heads and tails goes to zero as the number of tosses approaches infinity.
If you have a program like Matlab, it’s easy to write a program to see this, and graph the results. If you don’t have a computer to do it for you, you can try playing a few rounds. Make the length of the game something like 30 flips, and keep track of the difference between the number of heads and the number of tails after those 30 flips. I have a program that runs through 20000 iterations of this game, and the odds of getting the same number of heads and tails is roughly 15% - not exactly betting odds. If you increase the length of the game to 300, the odds drop to around 5%.
Everyone always starts off coin problems with the assumption that the coin is a fair coin. Is there really such a thing? Most coins are different on either side. Does anyone have any data on whether this significantly affects the probability of getting a tails or a heads? I read something about casinos using dice that don’t have drilled pips for that very reason- that it can make one outcome slightly more favorable than another.
Here’s another example: Suppose I’m flipping a fair coin 100 times. Before I make any flips, my expectation of the most probable distribution is 50 heads, 50 tails. Now, suppose I flip the coin the first 51 times, and they all happen to come up heads. What are my chances of getting a 50/50 split now?
“There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe”
–A. Einstein
I’m not sure what connection you see between these two situations. In the coin situation. you’re trying to predict the future from the past. In the car example, you’re figuring out what happened in the past by looking at the present.
As for your OP:
First of all, the probability that the number of heads will be between 48 and 52 is actually quite a bit lower than than 99%.
Secondly, it’s not a matter of combining long term with short term. The “long term” of which you speak includes all possible combinations of 100 flips. But by the time you have actually performed 10 flips, the vast majority of those possibilities no longer exist, and so it is flawed to base any conclusion upon htose possibilities that no longer exist. When you started, you didn’t know that the those possibilities wouldn’t occur, and so at the very beginning they exist as possibilities, and so they are relevant. But after ten tosses, the possibilities that no longer exist are not relevant.
