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The Coin
Before delving into the probabilities of multiple coin tosses, it is important to determine whether the ratio of heads to tails is truly a 50-50 proposition. In other words, determine if the distribution of weight in the coin skewed in a way that can affect the probability. According to Wolfram's Mathworld, a spinning penny lands on heads only 30 percent of the time. You have to test a coin to see if it is truly balanced, for if you toss it just three or four times, the results will not be statistically accurate.
Testing the Coin
To mathematicians, determining the accuracy of a coin toss is known as the relative frequency estimate of a probability. If you have an activity with two possible outcomes, how many times do you need to perform the task to prove that each possible result has an equal chance? For instance, if you flipped a coin four times, your chance of getting two heads and two tails would be 25 percent. By time the number of flips reaches 10,000, the probability of getting equal numbers of heads and tails falls very close to 50 percent.
Gambler's Ruin
In corollary to the above probability, it can be proven that it pays to gamble with a lot of coins. For instance if a gambler with a small number of chips plays against an institution with a large number of coins, the odds are in the favor of the bigger player, when the odds of each coin toss is 50:50. In other words the longer you play, the less likely there will be a deviation from the standard odds.
Saint Petersburg Paradox
The Saint Petersburg Paradox shows how doubling your wager after each bet will not work if the probability is 50 percent. In fact, you will only win the amount of your bet. For,example, on the first coin toss you gamble $2 that the coin will be heads. Heads you win $2, tails you lose and bet again at the $4 level. If heads comes up you will $4 after losing $2, so your net gain is $2. No matter how many times it takes to get the first heads, the net gain will only be $2.