1. Fivethirtyeight.com has opened its prediction data for everyone. Their NFL probabilistic predictions can be found here: https://projects.fivethirtyeight.comfl-apifl_elo.csv Calculate the Brier score, the accuracy (assuming a win probability threshold of 0.5) and the reliability curve for the predictions for the NFL seasons between 2010-2018. For the Brier score decompose it to its 3 parts (30 points). 2. An online shopping retailer is experimenting with two different versions of its website. The webpage randomly loads one of the two versions to every new request for the webpage. Version A was viewed by 3,250 customers from whom 1,320 ending up buying a product of the website. Version B was viewed by 2,870 customers from whom 1,250 purchased a product. The retailer is asking you whether any of the two version is associated with higher sales conversion (i.e., higher chance of a customer purchasing). What would you answer to them? (30 points) Hint: You can draw samples from a binomial distribution using the function random.binomial() from the numpy library in python. 3. Your professor told you that there are two slot machines (M1 and M2) in the Three Rivers Casino. One pays out 15% of the time, while the other 5% times. The only problem is that he cannot tell you which one is which, since he wants you to figure it out yourself based on what you learned in INFSCI 1530/2160. Next day you go to the Three Rivers Casino and chose one of the two machines randomly (say M1) and you put a coin. It does not pay out. What is your estimate of the probability that M1 is the machine you are

interested in? What if it had paid out - what would be your new estimate then? (40 points) Hint: What is the single most important rule in probability theory?

Data:

The data snippet is shown below and can be downloaded via the reference link above in question 1, only question 1 required the data information: