Statistical Calculators

Coinflip Probability Calculator

Use this free tool to figure out the odds of a coin flip.

coinflip probability calculator

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Table of contents

Tossing a Coin
Is a coin toss really fair?
Coin tossing is the technique
A Study on Coin Tosses


Simple explanation: how likely is for something to happen.
Many events cannot be predicted with 100% certainty. We can only predict how likely they will occur using the concept of probability.
In general, probability of an event equals the number of possible outcomes.
Classical probabilities problems often require you to find out how often one outcome happens versus the other and how future events will affect that outcome. The formula, which is similar to our coin flip probability formula, states that you should consider all possible outcomes.
Probability = number of successful results / number of possible results

Tossing a Coin

There are two outcomes possible when a coin is tossed:
  • Heads (H)
  • Tails (T)
  • The probability of the coin landing H is 1/2, and the probability of the coin landing T is also 1/2.

    Is a coin toss really fair?

    Since the beginning of time, people have relied on a simple and well-known technique to arrive at a decision free from biases or judgments. This method doesn't require complex machines to produce a result.
    To resolve an indecision, the most reliable method is to use some spare change and to toss a coin.

    Coin tossing is the technique

    It doesn't really need to be explained, since you have probably done at most a few of these before. But let's briefly outline what you should do when you toss your coin. Coin flipping, also known as coin tossing, involves throwing a coin in the air and choosing one of two outcomes: heads or tails. In order to reach a conclusion, each person involved in a decision chooses one of the sides. This is a common way to reach a decision (or resolve a disagreement). It is free of bias and both possible outcomes (heads and tails) are considered equally likely to occur.
    Coin tossing is a common activity in daily life. However, it is also popular on official and international platforms. The result of a coin toss was regarded as an expression of divine will in ancient times. The game of coin tossing was referred to by the Romans  naviga aut ("ship or the head"), while the British called the event cross and pile. Because a head or tail is the body part located at opposite ends of a person's body, it was often called 'heads' or 'tails'.
    These even odds may change in regular coin tosses where normal coins are used. First, new coins almost always come with imperfections and fabrications that can affect the coin's geometry. This is just one aspect. However, it is important to think about how frequently a coin changes hands. Coins can become damaged, worn-out, or scratched during the course of their lifetime. They can even hold bacteria. These factors can, however, remove fairness at a very limited level.
    There is another way to manipulate the odds. It is possible to train your fingers to flip the coin so that only the desired face appears every time you toss it. You may feel like you are having a bad luck day, but your opponent might be trained to accurately toss coins. They will take your money and, more importantly, your chances of winning a fair game.

    A Study on Coin Tosses

    The fairness of a coin-toss was determined by PersiDiaconis, Susan Holmes and Richard Montgomery. These results will completely change your perspective.
    The experiment used motion-capture cameras and random experimentation. A "coin flipper" was also used to flip the coin at will.
    These are some of their observations and inferences:
  • The coin will have a 51% chance to land on the same face it was launched from if it is tossed and caught. If it begins as heads, it has a 51% chance of ending as heads.
  • The chance that the coin ends with the heavier side down can be much greater if it is spun than tossed. Spun coins may show "huge bias", which can cause them to fall tails-up up to 80% of the times. To put it another way, you can only toss if you want fair play.
  • Tossing the coin and allowing it to clatter on the floor will add some randomness to the game. To promote fairness, allow the coin to clatter on a floor surface before picking it up.
  • The above mentioned spinning bias is likely to be present if the coin is tossed and left to tumble to the ground.
  • The same coin flip results can be produced by the same coin-flipping conditions. Certain conditions can produce the exact same result repeatedly, which is why there's some determinism to the coin-flip.
  • A stronger coin toss (more revolutions), decreases the bias.
  • A fascinating observation was that 1 out of every 6000 coin throws will result in a coin landing on its edge, creating a coin-flipping singularity. This is possible, and it was quite famously observed on December 8, 2013, during an NFL game between the Philadelphia Eagles (NFL) and the Detroit Lions.
    What are your thoughts? Is it still a good idea to toss a coin to determine who will pay for lunch?
    Parmis Kazemi
    Article author
    Parmis Kazemi
    Parmis is a content creator who has a passion for writing and creating new things. She is also highly interested in tech and enjoys learning new things.
    Coinflip Probability Calculator English
    Published: Thu Jul 21 2022
    In category Statistical calculators
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