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March 21st, 2011, 01:32 AM   #1
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Coin flipping game, approximation of one milion flips

Hi everybody,

first of all, i am not mathematician so my question is definitely quite noobish and my language will not be mathematical. I appologize

Long story short.

I am poker player and have written some blog posts (in one slovak website) about basic poker concepts. To explain situation with positive expected value (+EV situation) i.e. situation when we invest money with positive (monetary) expectation in longrun I used following example:

We will play the coin flipping game with following rules. If head falls, I loose 1 euro, if tail falls I gain 1,5 euro. We will make 1 000 000 flips with "fair" (not biased) coin.

As probability of head/tail falling is 0,5:0,5 we can calculate expected value of every flip as +0,25 euro. Further I stated that after 1 000 000 flips, we would see APPROXIMATELY the same number of heads and tails, actually I said something like "For the sake of this article lets neglect standard deviation" and made further calculation of expected value asi if 500 000 heads and 500 000 tails had fallen. I supposed that it was possible to make this approximation (of number of heads/tails) as after 1 000 000 flips the number of heads and tailes will be "close enough" to 500 000 heads/tails.

In discussion below this article some argument has arisen, when one guy stated, that we cannot neglect standard deviation as we will "never" see exactly 500 000 heads/tails (I know that) and siad something like that this kind of approximations have not been used since 80s (?). He used example of number line - we start at 0 and if head falls we move one part to the left (to "minus" part) if tail falls we move one part to the right (to "plus" part) - and after many flipps we will be further and further from 0 point. I think this is definitely true, but my point of view is following. The absolute value (distance from 0 point) will become greater and greater, but ratio of number of flips and number of heads/tails will be closer and closer to 1/2. And Imo if we apply this example to our coin flipping game we should move one part to the left if head falls, and 1,5 part to the right if tail falls.

The point is.:

Was my approximation correct, i.e. could I use this approximation when calculating expected value or was I wrong? (I am quite confused right now)
How can I calculate the probability of ending up in "loss" (or probabilty that my income will be geater than x or less than x) after 1 000 000 flips?

I will be really happy if somebody stops by and finds some time to answer.

Thx
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March 21st, 2011, 02:15 AM   #2
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Re: Coin flipping game, approximation of one milion flips

The expected value is certainly correct - i.e. if you were to perform the million-coin-toss experiment a large number of times then the mean number of heads recorded would approach 500,000.

As for the average difference between the recorded result and the 'perfect' 500k/500k split, from memory (you should check) it will be of the order of or 1000. In other words, if you think of your experiment in the number line sense the average mean distance away from the origin would approach something on the order of 1000.

However, since this is a simple binomial distribution with a very large N you will get a good approximation by the Normal approximation with mean 500000 and variance 250000. Using this to calculate the probability of returning a loss (no. of tails < 400000) gives a result more unlikely than finding Elvis riding the Loch Ness Monster.
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March 21st, 2011, 02:26 AM   #3
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Re: Coin flipping game, approximation of one milion flips

thanks man (bow) really appreciate
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March 21st, 2011, 06:10 AM   #4
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Re: Coin flipping game, approximation of one milion flips

Quote:
Originally Posted by mr_kaktus
The absolute value (distance from 0 point) will become greater and greater, but ratio of number of flips and number of heads/tails will be closer and closer to 1/2.
It's worth mentioning that, if the coin is truly fair, you expect the distance from 0 to decrease to 0 or 1 infinitely often, even as the expected value of the absolute distance from 0 increases.
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