Monte Carlo Lab Assignment

The program source code for each question must be submitted with a report which describes the problem and how your code works to solve the problem and the answer it calculates.

1. You are rolling two D12 dice (die with 12 sides with numbers 1-12).

a) Calculate the probability rolling two sixes.

b) Calculate the probability of rolling consecutive two sixes.

2. Consider the function:

f(θ) = sin(θ)/θ

a) Calculate the integral of f(θ) between 0 and 2π using the rejection method.

b) Calculate the integral of f(θ) between 0 and 2π using the evaluation method.

For both of your integral estimations provide an estimate of the standard deviation and standard error (using one sigma (68%) as the confidence interval) using the method introduced in Lab2.

c) Compare the standard deviation and standard error when generating 1x10³, 1x10⁴ and 1x10⁵ random numbers and explain the reason for this trend.

3. Robbie R. has stolen 1E4 debit cards and plans to start up a nest egg by withdrawing cash from ATMS. Each card will be frozen after four incorrect pin attempts. Based on pin number frequencies Robbie has decided that his first pin guess will be 1234 then 1111, 0000 and finally 1212, with the frequency of each PIN being 10.7, 6, 1.9, 1.2%, respectively. Robbie tries this 10 times (as in 10 groups of 1x104cards).

What is the average number of cards Robbie gains access to for each collection of 1x104cards? Assume that the rest of the PINS (besides 1234, 1111, 0000, 1212) all have an equal probability of being used.

You don't need to worry about leading zeros in a PIN. IE generating a random number of 12 and treating it as a PIN of 0012 is fine.

4. An exotic particle, the bobbyon, and its accompanying anti-particle, the anti-bobbyon, are located in a sphere with a radius of 10 mm. The bobbyon is initially located at (-1, 0,0)mm and the anti- bobbyon is located at (1, 0,0)mm if the two particles come within a radius of 1mm of one another they will annihilate.

Calculate the probability that the two particles will annihilate instead of leaving the sphere (average over about 1x105runs), assume the particles to be points.

Both particles have the same distribution of step size which varies uniformly between 0 and 1mm and at each step the particles scatters in a random direction isotropically.