Probability Your friend has a game night every week. For fun, he uses a fake 6-sided die on certain nights with probability P(D = f), otherwise, he uses the real 6-sided die. In this question, D = f denotes a fake die and D=-f denotes a real die. Re {1, 2, 3, 4, 5, 6} denotes the result of a roll. For example R= 1 means that a 1 was rolled. (a) Suppose you know the probability of rolling a 1, 2, 3, 4, and 5 given that the die is fake. In other words, you know P(R = 1|D = f), P(R= 2|D = f), P(R= 3|D= f), P(R= 4|D = f), and P(R= 5|D = f). Given this information, how would you find P(R= 6|D= f)? P(R = 6|D = f) = (b) This night, you are the first to roll. What is the probability that you roll a 6 (i.e. P(R = 6))? Use the product rule and marginalization to give your answer in terms of P(R = 6|D = f), P(R = 6|D = -).P(D = f), and P(D =-f). P(R = 6) = (c) Say you rolled a 6. The next turn, your friend rolls a 1. Given that you rolled a 6 and your friend rolled a 1, what is the probability of the die being fake (i.e. P(D= f\R1 =6, R2 = 1))? Use Baye's Rule to give your answer in terms of P(R1 = 6, R2 = 1|D = f), P(D= f), and P(R1 = 6, R2 = 1). %3D P(D = f|R1 = 6, R2 = 1) =

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Average

In mathematics, everyone must have heard the term average in one place or other. Almost all types of calculation require taking out an average. In everyday life, the term is used for denoting a certain amount that can be said to be a typical quantity for multiple people or objects. In other words, if a group of people possesses a certain object, then the average gives us an idea regarding the quantity of the object possessed by each person. However, this is solely based on mathematical calculation. It is important to note that most people will not strictly have the average value in a group.