Many people fervently believe that coin tosses are random.
The Indian men’s test cricket team captain Shubman Gill lost all five of his tosses against England this summer. The probability of such an event is only 3.1%, or one in 32, if winning or losing is assumed to be random. Some cricket enthusiasts realised then that Indian captains had thus lost 15 consecutive tosses in international cricket. Its probability, again assuming randomness, is a mere 0.003%, or one in 32,768. It’s extremely improbable to happen in one’s lifetime — and yet it did.
India commemorated the 50th anniversary of the Hindi film Sholay in August. One particularly memorable feature of the film was a double-headed coin. Amitabh Bachchan’s character Jai flipped this coin three times to make crucial decisions. Veeru, Dharmendra’s character, discovered at the very end that Jai was actually controlling all the choices he made because the coin always came up the same way.
The idea of using a coin to make decisions in Sholay was most likely inspired by the 1954 Hollywood film Garden of Evil, directed by Henry Hathaway. Richard Widmark played a gambler while Gary Cooper was a former lawyer. To determine who would stay back to fight the Apaches, Cooper and Widmark take turns to draw cards from a stack. Widmark eventually wins and stays back, allowing Cooper to ride to safety.
The coin in Sholay was a cultural revolution. In the late 1980s, when I was a student of statistics, it was the subject of a discussion in our class on probability — and remains so to this day.
A number of other films have used biased coins as well. Consider the 2007 Hollywood film No Country for Old Men. Its antagonist Anton Chigurh, played by celebrated actor Javier Bardem, is a menacing assassin who murders innocent victims, yet before each incident stops to flip a coin as if to give them one final hope. In one scene, Chigurh approaches a gas station owner and asks him to call head or tail on a toss. However, the coin settles upright on the edge before the owner can speak. Chigurh’s conviction is that chance determines one’s fate and the coin becomes a representation of his complex, warped ideology. It’s not about heads or tails at all.
Frank Capra directed the 1939 political comedy Mr. Smith Goes to Washington. A man named Mr. Smith travels to Washington only because a governor is attempting to choose between Mr. Hill and Mr. Miller, two competitive contenders, to be a senator. The governor flips a coin to select a candidate and it lands on the edge, so he picks Mr. Smith instead.
There were also a number of interesting coin tosses in the popular American animated sitcom ‘The Simpsons’. In the episode ‘The Monkey Suit’, a sports coach is fired for using a magnet to toss a coin. In one ‘Donald Duck’ comic from 1953, making all decisions by “flipping” a coin is called “flipism”. One peculiar fellow named Professor Batty advises the titular Duck to believe the coin. Donald does but soon finds himself in danger as a result.
In his 2007 book, Flooded by Randomness: The Hidden Role of Chance in Life and in the Markets, the essayist Nassim Nicholas Taleb wrote, “No matter how sophisticated our choices, how good we are at dominating the odds, randomness will have the last word.” The book described how human error, risk, probability, luck, uncertainty, and decision-making all work together to influence our actions.
But sometimes some events are more likely than others for reasons we can’t discern right away. According to the theories of 19th century physicists James Maxwell and Ludwig Boltzmann, randomness doesn’t equate to total unpredictability. When confronted with what the rules of quantum physics say about reality, Albert Einstein famously said, “God does not play dice with the universe” — an assertion that has been interpreted in many ways. Later, in his interpretation of the quantum world, Stephen Hawking stated, “Not only does God play dice but … he sometimes throws them where they cannot be seen.”
Coins that are tossed naturally follow the rules of classical mechanics and their initial conditions determine their flight through the air and the way they eventually land. In one interesting paper in 2007 titled ‘Dynamical Bias in the Coin Toss’, published in Siam Review, Persi Diaconis of Stanford University and his coauthors examined the natural process of flipping a coin that is eventually caught in one hand. They demonstrated that coins that are flipped with vigour tend to come up the same way they started. They also found the angle between a line perpendicular to the coin’s face and the vector of the coin’s angular momentum determined the limiting probability of the coin emerging this way. According to the authors, for natural flips, there is about a 51% chance of the coin coming up as started.
In the final analysis, if someone flips a biased coin and the other person calls head or tail in a truly 50/50 manner, the outcome (win/loss) will still be random. This can be demonstrated using basic calculations — and it’s why perhaps nobody can be blamed for the tosses the cricket captains lose. Then again, would we really want everything in our lives to be random?
Atanu Biswas is professor of statistics, Indian Statistical Institute, Kolkata.
Published – October 21, 2025 08:30 am IST
