Coin tossing question

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It is common knowledge that a coin toss has a 50/50 outcome and that if you tossed a coin 10 times you would get 5 heads and 5 tails. This can't be right, surely? For each individual toss, yes the probability is 50/50, but it is quite possible to toss a coin 10 times and get 10 tails. Why don't the results of multiple tosses reflect the probabilities of individual tosses?

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Last edited by fibonaccisequence94 on 25 Sep 2011, 7:42 am, edited 1 time in total.

The most likely outcome is 5 heads and 5 tails, the least likely (1 in 2^10, or 1 in 1024) is either all heads or all tails.

Take a look at the quincunx model, which explains the binomial distribution http://en.wikipedia.org/wiki/Bean_machine

What happens when you flip the coin multiple times is that you are multiplying the probabilities together. Any given outcome has a 1/2^n (one half to the nth power) chance of occurring, where n is the number of times you flip. However, you could probably guess that there you are more likely to get half heads and half tails than all heads or all tails. This is because there are more different ways you can get half and half than there are to get all one. (HHHTTHHTTT, HTHTHTHTHT, HHHHHTTTTT, etc. vs. HHHHHHHHHH only) The number of ways to get r heads in n flips (and therefore n-r tails) is given by nCr (n choose r), or n!/(r!(n-r)!). (n factorial divided by r factorial times n-r factorial.) Then divide that by 2^n (two to the nth power) to get the probability for that many heads.

"Would get" in the sense of "definitely" is definitely wrong. What is correct is that the EXPECTED VALUE of the number of heads is 5, leaving 5 tails. You may get 4, or 6, or 3, or even 10 one time in 1,024... but if the experiment of flipping a coin 10 times were run many times, the average should be somewhere near 5 heads and 5 tails.

You have to reckon whether multiple events are independent or not.

If someone tossed a million heads in a row (possible, but highly improbable) one would begin to wonder if the coin was fair or not.

ruveyn

1 in 9.9006562292958982507E+301029.... Pretty damn improbable.

Based on two assumptions:

1. The coin is fair
2. The tosses are independent events

ruveyn

If you actually got that, any statistician would conclude that the coin is not fair, precisely because the probability of getting it if the coin was fair is so tiny.

With such a large sample, the distribution of numbers of heads can be very closely approximated by a normal distribution with mean np and variance np(1-p). Variance is the square of the standard deviation, so the standard deviation of numbers of heads out of 1,000,000 is the square root of 250,000, or 500. Five hundred! Out of a million!! To get all one million of them as heads, that would be an incredible 1,000 standard deviations away from the mean!! ! Most of the time, you would reject a null hypothesis that the coin was fair with just 3 standard deviations away.

Laplace (Yes, that Laplace):

*Note: when he wrote divided, he meant "scattered and sparse", as with "united we stand, divided we fall". Yes, I'm also slightly mystified by his judicious use of the comma.


More Laplace:



Remember that the probability of getting a million heads in a row is the same as the probability of getting any other specific combination of a million heads and tails, the only difference for a million heads in a row is the human semantic impression that it is somehow "special".

In the language example: when we see the words Constantinople on the table, we presuppose that somebody must have intentionally arranged them as such, rather than the word appearing out of a random shuffling: simply because the word Constantinople is in the popular English language. Had the word Xpiayoc appeared, we would be more convinced that this is a nonsense word generated by random shuffling, although somebody from the ancient Mayan civilization would disagree:
http://en.wikipedia.org/wiki/Mayan_gods

No. It is possible to throw 10 heads in a row with a fair coin. The probability of doing so is 1/1024

ruveyn

I just love how people respond to the most trivial mathematical questions and stuff the thread with philosophy. That surely helps, doesn't it To OP - check if you understand what probability is, just a basic definition. There are many lovely popular books written on it!

there is a saying amongst mathematics teachers/professors: tell the truth, nothing but the truth, but never the whole truth.