Math 130A Probability Practice Midterm PDF

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UCI Fall 2020 Math 130A Practice midterm...

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PRACTICE TEST FOR MIDTERM

1. Combinatorial Analysis You have 10 identical coins that can be distributed into 15 different boxes. How many different possibilities are there if a) You can place at most one coin in each box? b) You can place at most two coins in each box? c) You can place as many coins in a box as you like? Hint: For part b), consider the different possibilities of how many coins are in a box with only one coin and how many will therefore be in boxes with two coins. 2. Probability Space You play a game against a stranger. Let p be the probability that the stranger wins if she does not cheat and 1 the probability that she wins if she cheats. Moreover, let q denote the probability that the stranger cheats. a) Given that the stranger won against you, what is the probability that she cheated? Show that this probability is larger than or equal to q . b) What happens for the two extreme values p = 0 and p = 1? Give a short interpretation of your result. 3. Knowledge is Power You are supposed to bet a large amount of money on the outcome of a single coin toss. However, you do not know the probability p with which the coin lands on “heads” (it could be any value in [0, 1]). But you are allowed to toss the coin one single time before you place your bet (a “practice toss”). In the following, we will discuss two different strategies on how to possibly increase your chances of winning. a) First strategy: you do not take advantage of your practice toss. Instead, when placing your bet, you just randomly pick (i.e. each with probability 1/2) “heads” or “tails”. What is your probability of winning when using this strategy? b) Second strategy: you do take advantage of your practice toss and decide to place your bet on the same outcome you observed for your practice toss. Show that your probability of winning using this strategy is given by (1 − 2p − 2p2 ). c) Let P(S1 ) and P(S2 ) denote the probabilities of winning, when using Strategies One and Two, respectively. Show that P(S1 ) ≤ P(S2 ). 4. The Prosecutor’s Fallacy A crime has been committed in a city with 10, 000 possible suspects. At the scene of the crime, the perpetrator left some traces of his DNA. Due to the nature of the evidence and the resources at the testing lab, the probability for a DNA match if the perpetrator is tested is 100%. However, the probability that the test shows an (incorrect) match if an innocent person is tested is 0.1%. Eager to solve this crime as quickly as possible, the district attorney now randomly picks one suspect after the other from the list and has their DNA tested. The first suspect to have a match will be charged with the crime (and no more testing will be done after this). What is the probability that an innocent person is indicted (assume that the test results of different suspects are independent)? 1

MIDTERM

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5. The St. Petersburg Paradoxon The following game is played:5.1 You keep flipping a fair coin until it lands on “tails”. a) Find a suitable sample space and an appropriate probability function to describe this random experiment. Now, once the random experiment is performed, let n denote the total number of coin tosses and you win 2n dollars. (For example, if the outcome is “HHT”, we have n = 3 and you win 23 = 8 dollars.) b) Let the random variable X denote your total winnings. What is E(X)? The reason, why this is called a “paradox”, is that according to the result in b), you should be willing to pay any finite amount of money to enter into this game. However, if you think about it, would you really be willing to bet everything you have so you can play this game? One suggested way to resolve this paradox is the observation that it is unrealistic, because one cannot win an arbitrary large amount of money. Let’s be generous and say you can win at most one trillion dollars. c) We still perform the same random experiment, however your total winnings will be the smaller of the two quantities: $2n or $1, 000, 000, 000, 000 (one trillion dollars). If the random variable Y denotes your winnings according to this new rule, what is E(Y )?

5.1This is a well-known problem and there are lot resources concerning this online. Since you are doing this exercise as preparation for the actual midterm, I strongly suggest you try to solve it on your own....