Subvert your cognition: the probability of flipping a coin is not 50%

Audit expert: Liu Yuhang

PhD, Beijing International Center for Mathematical Research

I believe many people have heard this sentence: "When you are faced with two different choices, tossing a coin always works, not because it will give the right choice, but because the moment the coin is tossed into the air, the answer you want will appear in your mind." The simple action of tossing a coin seems to assist people in making decisions by waiting for possible random results. In fact, in daily life, it is already inextricably linked to our emotions and psychology.

Source: pexels

A hundred years ago, mathematician Jacob Bernoulli proposed the famous "law of large numbers" through experimental verification and statistical analysis, and concluded that every time a coin is thrown upward and falls, the probability of the front or the back facing up is equal, both 50%. This conclusion was written into textbooks and has influenced generations of students. But do you know that the 50% probability can only be achieved under ideal conditions?

In other words, from the perspective of probability statistics, coin tossing is considered an ideal lottery process, which is not affected by factors such as the angle at which the researcher tosses the coin, the initial speed and direction of the coin, the shape, size, and quality of the coin. However, coin tossing in real life is a complex process, and its results are affected by the psychological and physiological factors of the tosser as well as the above factors. The existence of these interferences will cause some deviations in the final statistical results, causing the frequency data of the heads (tails) facing up obtained from the experiment to fluctuate around 50%.

Jacob Bernoulli Source: Baidu Encyclopedia

Coin Toss Model

In 2007, a physical model of human coin tossing developed by three researchers, Diaconis, Holmes, and Montgomery, suggested that the probability of an ordinary coin landing on the same side it originally landed on after being tossed is about 51%.

In the study, the researchers collected 46 different coins and conducted 350,757 tests. The experimental data was consistent with the model prediction. The probability of a coin landing on the same side, Pr (same side) = 0.50895%, with a credible interval (CI) of [0.506, 0.509].

However, other data show that the degree of deviation varies from person to person. Some people have a 60.1% chance of tossing the same side of the coin, while others have only a 48.7% chance. After analysis and research, it is believed that different people may produce different off-axis rotations when tossing a coin, causing the coin to shake, resulting in a higher same-side difference, but the probability of the coin falling on the same side at the beginning is still higher. Therefore, the research team concluded that when tossing an ordinary coin, it is more likely to land on the same side as the original, about 51%.

Source: pixabay

Rock, Paper, Scissors

Coincidentally, in real life, in addition to the unequal probabilities of the two sides facing up when tossing a coin, the winning rate of the well-known "Rock, Scissors, Paper" game is not fair at 1/3. According to data from a large number of human-computer experiments, the probabilities of people in a normal mental state making three gestures are rock (35.4%), scissors (35%), and cloth (29.6%). And through experiments, it can be observed that people who lose continuously or have obvious emotional fluctuations in the competition are more inclined to imitate the winner's last choice. Under the influence of human psychological factors, experimental results often deviate from theoretical values. In the closed-eye rock-paper-scissors experiment conducted on volunteers, the winning rate of players with open eyes was much lower than that of players with closed eyes. These experiments prove that complex and inaccurate psychological factors and the behavioral interference of the other party will affect the brain judgment of the participants, causing them to react differently in the game, resulting in the final experimental data not being the possible probability of winning with three gestures.

Rock Paper Scissors Game Source: Pixabay Dice Roll

In addition, the dice game that is common in life is also a game that seems fair, but is actually affected by multiple factors, and the probability of each result is not equal.

From a probability perspective, the number of points obtained from a dice throw should be random and equally likely, but in practice, factors such as the force used when throwing the dice, the angle at which the dice is thrown, the height from the table, the shape and size of the dice, and whether the mass is uniform will affect the final result. These factors combined result in the fact that the number of points cannot be equally likely when the dice is actually thrown, and it is also related to the number of points facing up when the dice is just thrown. Similarly, games such as throwing bottle caps also seem fair, but in fact, they are affected by multiple factors, and the results cannot satisfy the ratio of both sides facing up being equal.

Dice rolling source 丨pixabay

Since a large number of experimental results in the actual process cannot get exactly 50% of the ratio, why is it that in mathematical probability statistics, the probability of the positive (negative) side facing up is equal and each accounts for half? Let's turn our attention back to the law of large numbers mentioned at the beginning. Mathematicians have found that when a certain experiment is repeated a large number of times, the frequency of each result will be infinitely close to the probability of the event and stabilize near this value. This reflects the statistical regularity followed by random phenomena in a large number of repeated experiments, and there is a kind of inevitability in chance. The experimental data mentioned in the article are all the results obtained through a large number of independent and identically distributed repeated events, but in actual operation, they are affected by many factors, so it seems that there will be some deviations from the final probability. But when it is possible to strictly limit the experimental conditions and eliminate the interference of related factors as much as possible, the results obtained by infinitely repeated experiments will definitely be infinitely close to the ideal probability of the event.