Okay, so I’m new to this board, but I’ve read almost every post so far and the discussion seems to have gotten caught up in series, sums, etc. and moved a bit off of the probability angle. So here’s an explanation, without using sums (except in the background of the theory.)
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(In this, a game represents a specific method of play, i.e. flip until heads comes up. A match will be a series of games. )
It should be clear that from a probability standpoint, the two games are exactly the same (We’re not talking about payoff yet.) You flip until you get heads.
The expectation (defined, at any rate) is the mean value of the probability distribution. (1) See below for more. It represents essentially what happens in an infinite match of games, and is useful to study, among other things, likely payoff. In this case, the probablity distribution is well-known and is called the Geometric Distribution, defined as :
Waiting time T until first success of an event in Bernoulli trials (2) with probability p.
The expected value or mean of this distribution is 1/p . In other words, for this particular game, you’ll have an expectation of 2 turns until you win. This isn’t particularly important, but I thought you might like to know that this much of the game is a known distribution. This is just to explain that much, and to make it clear that the two games, if played infinitely without a particular payoff, will have to end up exactly the same.
Now to go to the payoff. It makes absolutely no difference to the finaly payoff if you pay off one game at a time, or just wait until the match is over, and figure the payoff. So for both of these, if you play a match of finite length, you’ll get some set of numbers at the end representing winning n, and pay off. If you used the same match numbers, you’d of course get twice the payoff using Game 2. But if the numbers are different (different matches) you can’t expect to get the same payoff, since you’ll have a different set of numbers.
Imagine a giant board (in fact it’s infinite) that has every single integer on it. Let’s say you put a marker on the board every time you play a game and reach that number. So after 10,000 games, you’d have a lot of markers near the bottom, and a few out over the other numbers. And if you have Board 1 and I have Board 2, they won’t necessarily look the same.
Now let’s keep playing (forever, in fact). It should be clear that as this happens, my board and your board will start to look pretty similar; in fact, at inifinite time, they must be equal. (3). All the spaces will be covered by markers, and the ones that are more covered will be the same. (If you aren’t seeing this, then review your concept of infinity and probability distributions). So if we pay off at infinite time (essentially what the OP was asking, since paying off after time and paying off as time passes are the same), we see that Board 2 will have twice the payoff of Board 1. Please follow up with any questions, but I have to admit I’m not that great at explaining this further.
panama jack
(1) The probability distribution is the way some random variable X looks in some range. It can be discrete or continuous, and has a value equal to the probability that X= some particular x in the range given. Ranges of infinite value are often considered. The normal distibution, or bell curve, is the most well-known probability distribution (for good reason, too, since every infinite set of distributions approaches it).
(2) This is mostly trivial, but Bernoulli trials are independent, and it must satisfy p+q=1 where q represents (not p occurs).
(3) If not, we can throw probability theory mostly out the window. The meaning of that statement, though, is that a probability distribution should work any time you use it, and represents essentially an infinite amount of events. There are in fact some who argue that we have to be careful about this, but it’s more philosophical. As an example of the problem, consider a weatherman’s prediction that there’s a 20% chance of rain tomorrow. Does this really have a concrete distribution. After all, there’s only one day that will ever be June 4, 2048 and so it’s difficult to say that this 20% represents a long term or infinite number of trials of June 4, 2048.
A few more notes :
Let’s look at the boards again. Imagine that instead of putting a marker on the number, we put a payoff on the number (like a casino game). It’s clear that the larger numbers will have huge sums stacked on them even after one win on them, while the low numbers will rack it up slowly. Still, one win at the large number will skew the running total by quite a lot. This is why computer simulation likely won’t work, and partly why some people think it doesn’t converge. The thing is, it does, but only at infinity. (It’s not really like a lot of things that do become bounded as time goes on.)