NPS Wk 9b - Monte Carlo Simulations

What is the Monty Hall Problem?

• Setup: A game show scenario with three doors; a car is behind one door and goats are behind the other two

• Initial choice: The contestant picks one door

• Host’s action: The host, knowing what is behind each door, opens one of the remaining two doors to reveal a goat

• Decision point: The contestant chooses to either stick with their original door, or switch to the other unopened door

• Perhaps somewhat counter-intuitively, the contestant should always switch in such a scenario

• One way to get a better sense of this is to literally map out all the possible cases

What is a state space?

The set of all possible configurations of a "system“

What is the birthday problem?

• The birthday problem is another well-known probability puzzle where the size of the state space does not pose a serious impediment to its solution

• The problem: Find the minimum value of 𝑘, such that in a group of 𝑘 people, it is more likely than not (probability > 50%) that there are least two people who share the same birthday

What is the Law of Large Numbers in statistics?

• The law states that the average of the results

  obtained from a large number of independent

  random samples converges to the true value

What is the idea Monte Carlo simulation?

Use randomness generated by a computer to perform a large number of virtual experiments, in order to estimate of some quantity of interest

Main steps in a Monte Carlo simulation?