Math in… Darts and Monte Carlo
If you throw darts at a board aiming for the bullseye, where do your misses land? You might expect them to land on the numbers 1 through 20 with about the same likelihood, but chances are there’s some bias.
Your throws might list right, for example, making 4, 13, 6, 10, and 15 more likely than 9, 14, 11, 8, and 16.
One way to figure out the bias is to make a lot of throws and see where they land. For example, if you make 1,000 throws and half of your misses hit 11, 8, 16, 7, 19, and 3, then you’d know your throws are listing to the lower left.
While you probably wouldn’t test this unless you’re a professional dart player looking to improve, Monte Carlo simulations can be used to approximate all sorts of probability calculations.
Suppose you play Probabilistic Tic Tac Toe. You start by randomly placing an X in one of the 9 spaces, each with equal likelihood. Then your opponent randomly places an O in one of the 8 remaining spaces, and so on, alternating until the game ends.
Who wins more often? To answer this question, we could map out all games and their associated probabilities. A far simpler way is to have a computer play this game over and over. In our computer simulation of 100,000,000 games, X won 58,486,423, O won 28,817,008, and 12,696,569 ended in a tie. It appears X wins about 58.5% of the time to O’s 28.8%.
The reason this method works is the law of large numbers. As the number of trials approaches infinity, the results of those trials conform to the underlying distribution.
It wouldn’t too suspicious to flip a coin 10 times and get 7 heads. However, if you flipped it 10,000 times and got 7,138 heads, you would be mathematically justified to suspect that the coin is unfairly weighted towards heads. As you increase the number of flips, you increase your accuracy in quantifying that bias.
Monte Carlo methods have applications to all sorts of problems, like network and circuit analysis, particle collision models, risk assessment, and climate forecasts.
What else could you investigate with these tools?
