Exploring Odds and Probability of a Symmetric Coin Toss

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SUMMARY

The probability of obtaining an even number of tails when tossing a symmetric coin 491 times is exactly 50%. This conclusion is derived from the principle of symmetry in probability, which states that the likelihood of getting n tails is equal to the likelihood of getting m tails, where n and m are the counts of tails and heads respectively, and n + m equals the total number of tosses. The mathematical proof relies on combinatorial arguments and the properties of binomial distributions.

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kaitoli
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Hi everyone

There is a question which I find very hard to solve and it goes like this..

A symmetric coin with heads on one side and tails on the other side is tossed 491 times after one another. The total amount of times you get tails is either even or odd. Is the probability that you get an even amount of tails exactly 50%? And the question requires a strong mathematical evidence, e.g. a formula.
 
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Mathematical proofs don't always require formulas. In this case, the basic argument is symmetry (assuming a fair coin). Therefore the probability of n tails and m heads is the same as the probability of m tails and n heads, where n and m are arbitrary with n+m=491.
 

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