Материал готовится,

пожалуйста, возвращайтесь позднее

пожалуйста, возвращайтесь позднее

Coin Tossing

Putting a new spin on randomness

In high school, I read Tom Stoppard’s 1967 play Rosencrantz and Guildenstern Are Dead, a hilarious take on the lives of two minor (and more or less interchangeable) characters from Hamlet. A lot of the dialog has to do with the philosophical question of destiny. At the beginning of the play, Rosencrantz and Guildenstern are tossing coins, and incredibly, 100 consecutive spins come up heads until a “lucky” toss finally comes up tails. This nicely illustrates the futility of the characters’ actions and also puts them squarely in some alternative reality — we all know that in the real world, coin tosses are random and couldn’t possibly come up heads 100 times in a row. We depend on this fact; otherwise, all the bets and disagreements that have been settled by this simple selection mechanism must be in doubt.

When I wrote about rock, paper, scissors tournaments, I made a passing reference to my favorite “binary random number generator,” a coin toss. A reader sent me a note saying that wasn’t quite accurate — coin tosses are not truly random. Talk about shaking the foundations of my faith. What insidious conspiracy could be behind this astonishing claim? Or could it simply be that a bunch of statisticians had entirely too much time on their hands?

To Err Is Human

Common sense tells us that a coin toss is random because we have no way to predict its outcome. We don’t know how many times a coin will spin in the air before landing, and we have no way of controlling the number of spins precisely; ergo, the outcome must be random. And casual observations bear this out: if you flip a coin 20 times, it will usually come up heads about half of the time. Random numbers being random, it may not be exactly 10 heads and 10 tails, but the larger your sample size, the closer you’ll get to a 50:50 ratio.

However, it turns out that when it comes to humans tossing coins, it really is a matter of uncertainty and inaccuracy rather than true randomness. Two Stanford University professors, mathematician Joseph Keller and statistician Persi Diaconis, have done extensive research and experimentation regarding coin tosses. Keller showed mathematically that the only way to obtain a truly random coin toss is to make sure it spins around its exact geometrical axis, something no human could do with the necessary precision. And even then, one would have to assume that the number of spins in any given toss was random. After all, coins, like everything else, must obey the laws of physics. Therefore, if you tossed a coin exactly the same way twice — taking into account velocity, angle, air resistance, and other physical variables — it would have to land exactly the same way both times. Diaconis had a machine built that does exactly that, and it can faithfully deliver 100 heads or 100 tails in a row by taking all the human variability out of the picture.

Of course, the effectiveness of the machine is based on the coin’s starting position. Given the number of spins its mechanism creates for every coin toss, the coin will end up on whichever side was up when it started. (Had the machine been designed to use a little more force or a little less, the coin would have ended up on the opposite side each time.) Diaconis wondered if even for humans with our built-in randomizing flaws, there might be a similar bias — a statistical likelihood that a given initial condition will make a given final condition occur more frequently.

Putting a new spin on randomness

In high school, I read Tom Stoppard’s 1967 play Rosencrantz and Guildenstern Are Dead, a hilarious take on the lives of two minor (and more or less interchangeable) characters from Hamlet. A lot of the dialog has to do with the philosophical question of destiny. At the beginning of the play, Rosencrantz and Guildenstern are tossing coins, and incredibly, 100 consecutive spins come up heads until a “lucky” toss finally comes up tails. This nicely illustrates the futility of the characters’ actions and also puts them squarely in some alternative reality — we all know that in the real world, coin tosses are random and couldn’t possibly come up heads 100 times in a row. We depend on this fact; otherwise, all the bets and disagreements that have been settled by this simple selection mechanism must be in doubt.

When I wrote about rock, paper, scissors tournaments, I made a passing reference to my favorite “binary random number generator,” a coin toss. A reader sent me a note saying that wasn’t quite accurate — coin tosses are not truly random. Talk about shaking the foundations of my faith. What insidious conspiracy could be behind this astonishing claim? Or could it simply be that a bunch of statisticians had entirely too much time on their hands?

To Err Is Human

Common sense tells us that a coin toss is random because we have no way to predict its outcome. We don’t know how many times a coin will spin in the air before landing, and we have no way of controlling the number of spins precisely; ergo, the outcome must be random. And casual observations bear this out: if you flip a coin 20 times, it will usually come up heads about half of the time. Random numbers being random, it may not be exactly 10 heads and 10 tails, but the larger your sample size, the closer you’ll get to a 50:50 ratio.

However, it turns out that when it comes to humans tossing coins, it really is a matter of uncertainty and inaccuracy rather than true randomness. Two Stanford University professors, mathematician Joseph Keller and statistician Persi Diaconis, have done extensive research and experimentation regarding coin tosses. Keller showed mathematically that the only way to obtain a truly random coin toss is to make sure it spins around its exact geometrical axis, something no human could do with the necessary precision. And even then, one would have to assume that the number of spins in any given toss was random. After all, coins, like everything else, must obey the laws of physics. Therefore, if you tossed a coin exactly the same way twice — taking into account velocity, angle, air resistance, and other physical variables — it would have to land exactly the same way both times. Diaconis had a machine built that does exactly that, and it can faithfully deliver 100 heads or 100 tails in a row by taking all the human variability out of the picture.

Of course, the effectiveness of the machine is based on the coin’s starting position. Given the number of spins its mechanism creates for every coin toss, the coin will end up on whichever side was up when it started. (Had the machine been designed to use a little more force or a little less, the coin would have ended up on the opposite side each time.) Diaconis wondered if even for humans with our built-in randomizing flaws, there might be a similar bias — a statistical likelihood that a given initial condition will make a given final condition occur more frequently.

Загрузка...

Выбрать следующее задание

Ты добавил

Выбрать следующее задание

Ты добавил