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Fixed some code/math

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Brandon Rozek 2022-01-12 23:45:54 -05:00
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content/blog/2017-03-14-monte-carlo-pi.md
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@ -62,9 +62,9 @@ To add up these small areas we need to make an assumption. The assumption is tha
This will allow us to perform a pooled empiricle probability on the simulations to sum up the areas.
Meaning the area of the circle will be the number of times that the inequality was satisfied $$A_{circle} = \# Successes$$
Meaning the area of the circle will be the number of times that the inequality was satisfied $$A_{circle} = \\# Successes$$
And the area of the square will be the number of times the simulation was run, since the random numbers generated will always be between 0 and 1 $A_{square} = \# Trials$
And the area of the square will be the number of times the simulation was run, since the random numbers generated will always be between 0 and 1 $A_{square} = \\# Trials$
Recall that taking the ratio of the area of the circle and the area of the square is a fourth of pi. $$\frac{\frac{1}{4} \pi}{1} = \frac{1}{4} \pi$$
@ -72,7 +72,7 @@ Multiply this number by 4 and you get the value for pi.
This tells us that four times the probability that the randomly generated point is in the circle is equal to pi.
$$\pi = 4 * (Probability\ of\ being\ inside\ circle) = 4 * \frac{\# Success}{\# Trials} = 4 * \frac{A\_{circle}}{A\_{square}}$$
$$\pi = 4 * (Probability\ of\ being\ inside\ circle) = 4 * \frac{\\# Success}{\\# Trials} = 4 * \frac{A\_{circle}}{A\_{square}}$$
## Implementation
@ -90,11 +90,12 @@ class MonteCarloPi {
BigInteger successes = BigInteger.ZERO;
BigInteger trials = BigInteger.ZERO;
</code></pre>
```
For this simulation, we will run 1,000,000,000 trials
<pre class='language-java'><code class='language-java'> BigInteger numTrials = new BigInteger("1000000000");
```java
BigInteger numTrials = new BigInteger("1000000000");
/*
Monte Carlo Simulation
Generate a random point 0 &lt;= x &lt; 1 and 0 &lt;= y &lt; 1