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If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure: [4] Toss the coin twice. If the results match, start over, forgetting both results. If the results differ, use the first result, forgetting the ...
A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both. In the coin-tossing example, both outcomes are, in theory, collectively exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities. [1]
While a run of five heads has a probability of 1 / 32 = 0.03125 (a little over 3%), the misunderstanding lies in not realizing that this is the case only before the first coin is tossed. After the first four tosses in this example, the results are no longer unknown, so their probabilities are at that point equal to 1 (100%).
Where the money goes. Some well-known fountains can collect thousands of dollars in coins each year. According to an NBC report from 2016, the Trevi Fountain accumulated about $1.5 million in ...
For example, if a typical coin is tossed and one assumes that it cannot land on its edge, then it can either land showing "heads" or "tails." Because these two outcomes are mutually exclusive (i.e. the coin cannot simultaneously show both heads and tails) and collectively exhaustive (i.e. there are no other possible outcomes not represented ...
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails) appears. William Feller showed [1] that if this probability is written as p(n,k) then
Five coins were confiscated and destroyed before they were acknowledged as genuine. Editor’s note: This article was updated to reflect the 1982 penny is a Lincoln penny, not a wheat penny.
Common intuition suggests that if a fair coin is tossed many times, then roughly half of the time it will turn up heads, and the other half it will turn up tails. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of heads to the number of tails will approach unity.