Tom and Jerry play a fair dice game. In a game, each of them rolls the dice once, the one who gets a larger number wins (tie is allowed, i.e., they get the same number). They play a total of n games. (a) Calculate the probability p that Tom wins for an independent game. (b) Let X be the number of games that Tom wins. What is the distribution of X? Write down your formula, give a few sentences to justify. (c) Let Y be the number of ties. What is the distribution of X+Y? Write down your formula, give a few sentences to justify.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Tom and Jerry play a fair dice game. In a game, each of them rolls the dice once, the one who gets a larger number wins (tie is allowed, i.e., they get the same number). They play a total of n games.
(a) Calculate the
(b) Let X be the number of games that Tom wins. What is the distribution of X? Write down your formula, give a few sentences to justify.
(c) Let Y be the number of ties. What is the distribution of X+Y? Write down your formula, give a few sentences to justify.
(d) Let P be the number of games that Tom wins in the first k games, Q be the number of games that Tom wins at the last r games. Are P and Q independent? Give your answer and a few sentences to justify.
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