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Maths Not Magic

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Flip Coin

Date Published: 18th April 2020

Author: Matt Clarke

Many people today might not understand the solemnity of coin flipping, after all it’s done casually among friends and families to break ties and make quick decisions. Yet throughout the history of sports, the results of flipping a coin caused such ripple effects in particular games, to the point of impacting wins and losses, (Mather, 2015).

It is unclear where and when coin flipping first became a determining method. But in ancient times the outcome of flipping a coin was often seen as the expression of a higher power’s will. According to Komonews, the first accounts were during the Roman Empire times when it was referred to as navia aut caput meaning ship or head. In England, it was cross and pile while in Peru, it was face or seal, basically, the symbols on either side of every coin.

Today some internationally recognized sporting events have incorporated coin flipping as a determiner for certain aspects of the games for example; The Super Bowl and The Cricket World Cup. However, (Weintraub, 2010) states that coin flipping has lost its primacy in professional sports. It is still done in many sports but not to determine wins and championships as in the past.

Coin flipping is now done to determine things like, which team gets to shoot or defend first and in what direction of the field. Which player serves first in games like tennis, which teams go first or second in penalty shootouts and which team begins overtime with possession of the ball. In short, coin flipping is relied on to fairly guide the nitty grittys of various games.

All this information then leads to the main question, are the odds of coin flipping an assured 50/50 chance? According to multiple sources online the short answer is NO. There is, at minimum, a one percent bias that may seem fairly insignificant. But the severity of this difference is proved in a conclusive study undertaken by three mathematicians and statisticians.

(Diaconis, Holmes & Montgomery, 2007) argue that every time a coin is tossed up, its chances of landing on the same side it was while being tossed are 51%. Meaning if the coin was heads up, 51 out of 49 times, it will remain heads up. An online article done by the Coding Wheel supports this statement by explaining it in terms of odd and even numbers.

Every time a person counts odd and even numbers from 1, there can only be two of these results. Either the number of odd numbers counted are more than the even or the number of odd numbers are equal to the even. The same applies to a coin, if the toss begins heads up, the amount of time that is counted on air will always be more or equal to the tail. In no way can the time spent on the tail side be more unless it began the toss facing upwards.

Another major factor lies in the asymmetry of some coins. Usually, the head side of a coin is heavier than the tail due to the embellishments on the faces of leaders. Resulting to the coin always falling on the heavier side when being spun. Even so, this doesn’t affect a coin that is being tossed on air. The spin bias comes in when the edge of a coin spins on a flat surface before it stops, Diaconis et al. (2007). Many people think that letting a coin land on the ground increases the randomness of the results but this is only true if the coin doesn’t spin before it stops.

There are more intense factors like air resistance that have been reviewed in the study by Diaconis et al. (2007). Having said all this though, there are some things that people can do to increase randomness of a flip. For example, before flipping a coin you should vigorously shake it between the palms of your hands to remove the throwing side bias. It’s also been proven that the more revolutions a coin makes on air, the more the bias is decreased.

If the person flipping the coin is aware of the factors that affect a coin toss, they would be able to get their desired outcome 10/10 times. As for sports tosses, the coins used are specifically made for that reason meaning that the weight bias can be eliminated but not the throw side bias. Therefore, it is safe to conclude that the odds of coin flipping are never 50/50 chance.

It is unclear where and when coin flipping first became a determining method. But in ancient times the outcome of flipping a coin was often seen as the expression of a higher power’s will. According to Komonews, the first accounts were during the Roman Empire times when it was referred to as navia aut caput meaning ship or head. In England, it was cross and pile while in Peru, it was face or seal, basically, the symbols on either side of every coin.

Today some internationally recognized sporting events have incorporated coin flipping as a determiner for certain aspects of the games for example; The Super Bowl and The Cricket World Cup. However, (Weintraub, 2010) states that coin flipping has lost its primacy in professional sports. It is still done in many sports but not to determine wins and championships as in the past.

Coin flipping is now done to determine things like, which team gets to shoot or defend first and in what direction of the field. Which player serves first in games like tennis, which teams go first or second in penalty shootouts and which team begins overtime with possession of the ball. In short, coin flipping is relied on to fairly guide the nitty grittys of various games.

All this information then leads to the main question, are the odds of coin flipping an assured 50/50 chance? According to multiple sources online the short answer is NO. There is, at minimum, a one percent bias that may seem fairly insignificant. But the severity of this difference is proved in a conclusive study undertaken by three mathematicians and statisticians.

(Diaconis, Holmes & Montgomery, 2007) argue that every time a coin is tossed up, its chances of landing on the same side it was while being tossed are 51%. Meaning if the coin was heads up, 51 out of 49 times, it will remain heads up. An online article done by the Coding Wheel supports this statement by explaining it in terms of odd and even numbers.

Every time a person counts odd and even numbers from 1, there can only be two of these results. Either the number of odd numbers counted are more than the even or the number of odd numbers are equal to the even. The same applies to a coin, if the toss begins heads up, the amount of time that is counted on air will always be more or equal to the tail. In no way can the time spent on the tail side be more unless it began the toss facing upwards.

Another major factor lies in the asymmetry of some coins. Usually, the head side of a coin is heavier than the tail due to the embellishments on the faces of leaders. Resulting to the coin always falling on the heavier side when being spun. Even so, this doesn’t affect a coin that is being tossed on air. The spin bias comes in when the edge of a coin spins on a flat surface before it stops, Diaconis et al. (2007). Many people think that letting a coin land on the ground increases the randomness of the results but this is only true if the coin doesn’t spin before it stops.

There are more intense factors like air resistance that have been reviewed in the study by Diaconis et al. (2007). Having said all this though, there are some things that people can do to increase randomness of a flip. For example, before flipping a coin you should vigorously shake it between the palms of your hands to remove the throwing side bias. It’s also been proven that the more revolutions a coin makes on air, the more the bias is decreased.

If the person flipping the coin is aware of the factors that affect a coin toss, they would be able to get their desired outcome 10/10 times. As for sports tosses, the coins used are specifically made for that reason meaning that the weight bias can be eliminated but not the throw side bias. Therefore, it is safe to conclude that the odds of coin flipping are never 50/50 chance.

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