# coin toss

The coin toss is the simplest real chance experiment . In the idealized case, it has two outcomes, heads or tails, whose probabilities are equal with 50% each. In fact, it is also possible for the coin to land on the edge. However, this is very rare. Depending on the coin, there is also a minimal imbalance due to the weight difference between the sides.

The random experiment is often used in sports, for example soccer or American football . The coin toss serves as a random mechanism in two-up , a game of chance offered in many Australian casinos . In the case of the fox , a coin has to be thrown at a certain place; however, this is a game of skill and not a game of chance.

In the card game poker , the term coin flip is used to describe a situation in which two players with roughly equally strong hands are fighting for a win.

## Application examples of real coin flips

• In a bet or in a game of chance
• In football , the referee tosses a coin to decide which team can choose their own half of the field and which can kick off. Before the introduction of the penalty shootout , the winner was sometimes determined in an elimination game if, depending on the competition rules, after overtime or overtime in a replay, no winner was determined.
• The best-known example of this is the " coin toss of Rotterdam ": On March 24, 1965, the entry into the semi-finals of the European Cup winners of the national champions in international football was crowned by a coin toss. After all 3 games (return and playoff plus extra time 0-0; 0-0; 2-2) between 1. FC Köln and Liverpool FC had ended in a draw and a penalty shoot-out was not yet planned, the Belgian threw Referee Schout a coin. On the first throw, however, it got stuck vertically in the ground. At the second litter she decided in favor of the English.
• In American football , the referee throws a coin (coin toss) , whereby the team captain can predict the result (heads or tails) for the away team. The team that wins the coin toss can decide whether to kickoff first, let the opponent kickoff first, choose the half to be defended first, or cancel this option for the second half of the game. The other team may choose one of the remaining options.

## Application in probability theory

The coin toss is often used in probability theory as a simple prototype of a random experiment. This experiment is described with the following model:

• ${\ displaystyle \ Omega = \ {K, Z \}}$ describes the possible outcomes of the experiment: The coin shows heads (K) or tails (Z).
• The probability distribution on is determined by.${\ displaystyle \ Omega}$ ${\ displaystyle P (\ {K \}) = P (\ {Z \}) = 50 \, \%}$ • The random variable is determined by and .${\ displaystyle X \ colon \ Omega \ to \ {0,1 \}}$ ${\ displaystyle X (K) = 0}$ ${\ displaystyle X (Z) = 1}$ This is a discrete, uniformly distributed random variable. ${\ displaystyle X}$ The model can also be varied as follows:

• If the coin is not ideal (but, for example, marked), a "head probability" between 0 and 1 is determined differently .${\ displaystyle P (\ {K \}) = p, \; P (\ {Z \}) = 1-p}$ ${\ displaystyle p}$ • The random variable is sometimes used with a more general definition and , where a and b are two (meaningfully different) real numbers.${\ displaystyle X}$ ${\ displaystyle X (K) = a}$ ${\ displaystyle X (Z) = b}$ 