 # The Coin Toss: Probabilities and Patterns

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Coin Tossing: The Hydrogen Atom of Probability

The coin toss is really just a metaphor for a random event that has only two possible outcomes. The actual tossing of a real coin is just one way to realize such an event. There are many examples of things that are equivalent to a coin toss:

• Will the stock market close up or down tomorrow?
• Will a die roll come up with an even or odd number?

• Will we make contact with extraterrestrials within the next ten years?
• Will a car drive by in the next minute?

• Will tomorrow be sunny or cloudy?

• Will my medical test result be negative or positive?
• Will I enjoy this movie?
• Will the next joke be funny?
• Will the Earth's average temperature go up next year?

Because the coin toss is the simplest random event you can imagine, many questions about coin tossing can be asked and answered in great depth. The simplicity of the coin toss also opens the road to more advanced probability theories dealing with events with an infinite number of possible outcomes.

With this book, you can answer questions like:

• Is it unusual to get only 6 heads in 20 tosses of a fair coin? (pg 12)
• In 100 tosses of a fair coin, what is the probability there will be no more than 40 heads? (pg 22)
• What is the probability that a fair coin has to be tossed 5 times before the first head appears? (pg 23)
• What is the probability that it takes 10 tosses to get 3 heads, with probability of heads=0.4? (pg 24)
• How is the coin toss related to the amount of time it takes for an unstable nucleus to decay? (pg 30)
• If you are betting on a coin toss whose probability of heads varies over time, what is the best strategy to use? (pg 33)
• How are the Catalan numbers related to coin tosses? (pg 45)
• How is the probability of losing all your money in a gambling game related to coin tosses? (pg 46)
• Playing an unfavorable game against an opponent with unlimited resources, what's the average time before you're broke? (pg 60)
• If you're tossing a coin once per second, and after about 40 minutes you suddenly get 10 heads in a row, should you be surprised? (pg 65)
• What is the probability that a run of length 5 heads occurs somewhere in the first 25 tosses of a fair coin? (pg 71)
• How are the Fibonacci numbers related to coin tosses? (pg 73)
• What is the probability that it takes 100 tosses to get more than one run of 5 heads with probability of heads = 0.49? (pg 83)
• What is the probability that on the 50th toss we get 5 heads in a row with a fair coin? (pg 88)
• What is the average number of tosses needed to get either 10 heads in a row or 10 tails in a row using a coin with probability of heads = 0.55? (pg 92)
• What is the probability that a run of 10 heads or 6 tails occurs for the first time on the 50th toss using a coin with probability of heads = 0.7? (pg 94)
• What is the probability that the pattern HTHTH occurs on toss 50 using a fair coin? (pg 104)
• For a coin with heads probability = 0.55, what is the probability that in a sequence of 50 tosses, a head never comes up more than 6 times in a row? (pg 105)
• For a fair coin, what is the probability that the longest run of heads or tails in a sequence of 30 tosses is less than or equal to 5? (pg 107)

This book is very mathematical. Some knowledge of calculus, discrete math, and generating functions is helpful to get the most out of it. A review of discrete math is provided in the index.

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