Problem 1: Random Dice (2pts)
This problem will compare the theoretical properties of a fair die to empirical results from simulation. It will further familiarize you with the numpy.random library.
a) Assume that we roll two fair six-sided dice. What is the probability of observing double sixes?
b) Let the random variable X represent the result of rolling two fair six-sided dice. For example X = (1, 5) is the event that we roll (1, 5). What distribution does X follow and what are the parameters of the distribution?
c) Initialize the random seed to 0 using numpy.random.seed. Using numpy.random.randint simulate 1,000 throws of two fair six-sided dice. From these simulations, what is the empirical probability of double sixes (the percentage of times this event occurred in simulation)?
d) Reset the random seed to 0 and repeat the above simulation a total of 10 times and report the empirical probability of double sixes for each run.
e) The empirical probability of double sixes from each simulation will differ. Yet, the probability of double sixes is fixed and was calculated in part (a) above. Why do these numbers disagree?
f ) In the above we flipped 1, 000 coins 10 times each. How would our results change if we flip 5, 000 coins 10 times each? How would they change if we flip 1, 000 coins 100 times each?
Problem 4: Conditional Probability (2pts)
a) Assume that we roll two fair six-sided dice. What is P(sum is 5 | first die is 3)? What is P(sum is 5 | first die is 1)?
b) Assume that we roll two fair four-sided dice. What is P(sum is at least 3)? What is P(First die is 1)? What is P(sum is at least 3 | first die is 1)?
c) Suppose two players each roll a die, and the one with the highest roll wins. Each roll is considered a “round” and further suppose that ties magically don’t happen (or those rounds are simply ignored) so there is always a winner. The best out of 7 rounds wins the match (in other words the first to win 4 rounds wins the match). Let W be the event that you win the whole match. Let S = (i, j) be the current score where you have i wins and the opponent has j wins. Determine the probability that you win the matchm, given the current score P(W | S = (i, j)) for each of the 16 possible values of S = (i, j).
To help you out a bit with part (c) above I am providing 4 big hints:
1) If the score is equal–that is S = (k, k)–then there is equal chance of either player winning the match, P(W | S = (k, k)) = 1 2 for k = 0, 1, 2, 3.
2) Let Ri be a random variable where Ri = 1 if you win round i and Ri = 0 if you lose that round. Note that P(Ri = 0) = P(Ri = 1) = 1 2 .
3) Recall that by the law of total probability P(W | S = (i, j)) = P(W, Ri+j+1 = 1 | S = (i, j)) + P(W, Ri+j+1 = 0 | S = (i, j)).
4) By the probability chain rule P(W, Ri+j+1 | S = (i, j)) = P(W | Ri+j+1, S = (i, j))P(Ri+j+1 | S = (i, j))
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