1. Monte Carlo Simulations
Steven Gollmer
Cedarville University
Picture from wikipedia - Dice

2. Meet and Greet Game Are there people here who share the same birthday?
Most births occur in September & October
October 5th is the most common birthday
May 22nd is the least common birthday
What does this have to do with Monte Carlo?
What is the probability that at least one pair of people share a birthday in this room?
Wikipedia – Birthday picture

3. Birthday Attack
Hash functions convert arbitrary length data into a fixed length string of data.
Passwords can be hashed for quicker verification.
Documents can be hashed and assigned to a digital signature.
Collisions occur when documents hash to the same value.
Image from wikipedia
Birthday attack information from
What impact does hashing have on security?
Cracking a password with a 64 bit hash with 100,000 attempts/second could take 5.8 Myr, but a collision with the hacker’s guess will likely occur in 14 hr.

4. Probability
Terms
Event (A) – The outcome of an experiment.
Sample space (W) – All possible outcomes for an experiment.
Probability – Ratio of |A| to |W |
What is the probability of rolling a seven on two dice?

5. Probability of Rolling 72 3 4 5 6 8 9 10 11 12
Chances are 1 out of 6

6. Birthday Problem
What is the probability of shared birthdays if there are N people in the room?
N = 1, P(1) = 0%
N = 365, P(365) = 100%
N = 3,
Choices
365 1
364 A A B
In General A B A
Combinations B A A

7. Additional Complications
What about 2 pairs of shared birthdays?
What about triple shared birthdays?
Are birthdays evenly distributed throughout the year?

8. Monte Carlo Simulation
Run multiple experiments, N0
Individual outcomes are either 0 or 1 (Failure or Success)
Total number of successes, Ns
Running more experiments give more accurate results.
Relative error
Probability

9. Calculate the value of p
Find the % of points falling within a circle of radius 1.
N0 = 10,000
N0 = 100,000
p =
p = ±
p =
p = ±

10. Monte Carlo – Birthday Problem
30 people in a room
Count when collisions occur
Probability theory – p =

11. Non-Uniform Birthday Problem
Theory – 70.63%
Monte Carlo
N=1k, p=70.4 ± 2.9%
N=10k, p=71.39 ± 0.90%
N=100k, p=70.90 ± 0.29%
N=1M, p=70.95 ± 0.09%
Normal Distribution
95% confidence interval 2s

12. Hypothesis Testing
Hypothesis: Does a non-uniform birthday distribution give a significantly different probability to the birthday problem?
Test: Ran 2 Monte Carlo simulations using a uniform and non-uniform birthday distribution.
Conclusion:
With a confidence level better than 95%, the hypothesis is verified. If a higher confidence is desired, use more experiments.

13. How well do climate models handle transport of sunlight?
Models assume that where clouds exist, they have uniform optical properties.
Marine Stratocumulus
Off coast of S. California
Representative of uniform clouds.
How do you test if this assumption is valid?

14. Fractal Clouds and Monte Carlo
Water content is redistributed using a fractal algorithm.
Send photons through the non-uniform cloud.
4096 bands
4 M photons
Albedo – % reflected
15% difference at 60% albedo
<1% smoothing from horizontal transport.

15. Other Applications Radiation Therapy Telescope design NOAA POES MEPED
Dosage from multiple photon scattering in tissue.
Telescope design
Impact of different configurations.
NOAA POES MEPED
Measure particle radiation that can disrupt communications.
Image from wikipedia on Monte Carlo photon simulation
NOAA-N from Nasa website

16. Conclusion – Monte Carlo Simulations
Advantages
Bypass the complexity of solving problems by analytical methods ( trigonometry, calculus, D.E., etc.)
Give solutions to problems too costly to compute using standard numerical methods.
Easily distributed to multiple processors or GPU’s
Disadvantages
Solution is statistical by nature.
High precision comes at a high computational cost.
Best used for problems with limited observables.

17. Credits www.r-project.org – R statistics program
NASA & NOAA – Image source
Wikipedia – General information and public commons images.
Cahalan et al Independent Pixel and Monte Carlo Estimates of Stratocumulus Albedo. J. Atmos. Sci. 51:
(1978 birthday data)