Define the random variable as, where is a randomly chosen number between and. This truly surprising outcome was discovered by American mathematician Kent Morrison.Įlementary probability is all you need to shed light on the result. In this way, the casino guarantees itself winning odds of at least, no matter what strategy the player applies. It uses this number to calculate a number and prints out the largest four-digit integer below on the casino’s slip of paper, generated at the start of the game. For every interaction, i.e., every time the game is played, the random-number generator on the casino’s computer picks a randomly chosen number between 0 and 1. Randomization is a technique that has many applications in mathematics and computer science. The trick is that the casino uses randomization in generating its four-digit integer. No matter what strategy the player uses, in the long run the casino wins at least 60.1% of the games. Unfortunately, this is, in fact, the expected outcome.
How could this happen? Your calculations were correct, and indicated a comfortable winning margin, but there you are, losing. After playing the game a great number of times, however, you are baffled to find yourself on the losing end of things. You hightail it to the nearest casino to claim your winnings. It certainly looks as though the casino has blundered, but hey, that is not your problem. This would mean that, in the long run, you will win on average dollar for every dollar staked, giving you on average a winning margin of slightly more than 5% for any four-digit integer chosen. Your computer program alerts you to the fact that 43.0% of the products begin with 4, 5, 6, 7, 8 or 9. Before deciding whether or not to play this game, you go off to calculate for yourself the possible products of two four-digit integers, from 1000 to 9999, in case you should play using a four-digit integer.
This sounds tempting, seems too good to be true. If the player wins, he gets 2.45 dollars for every dollar staked. The player wins if the product of the two integers begins with a 4, 5, 6, 7, 8 or 9 otherwise, the casino wins. The player’s integer is then multiplied together with the four-digit casino integer. The player must choose a positive integer having as many digits as the player wishes.
The player may not see this integer until he chooses one for himself.
The employee presses a button to generate a slip of paper printed with a four-digit integer on its reverse side. The game is played at a table between a player and a casino employee. You’re curious, so you go into the casino to find out what the rules of the game are. You’re strolling past a casino and you notice an eye-grabbing sign:
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