# Statistical Analysis of Engineering Systems Homework 4

This homework addresses the material covered in Lectures 3 and 4 on Chapter 1 in the text [1].

Recommended supplemental problems from text. These problems are not to be handed in with your assignment. You are encouraged to work with your classmates on the supplemental problems. • §1.4. Total probability theorem and Bayes’ rule: Problems 20, 21, 22, 23, 27. • §1.5. Independence: Problems 40, 41. • §1.6. Counting: Problems 51, 53, 59. Required problems for homework: 1. Alice searches for her term paper in her ﬁling cabinet which has several drawers. She knows that she left her term paper in drawer j with probability pj 0. The drawers are so messy that even if she correctly guesses that the term paper is in drawer i, the probability that she ﬁnds it is only di. Alice searches in a particular drawer, say drawer i, but the search is unsuccessful. Conditioned on this event, show that the probability that her paper is in drawer j is given by pj 1−pidi, j 6= i, pi(1−di) 1−pidi , j = i. (1) Hint: deﬁne events Ij (term paper is in drawer j), Lj (Alice looks in drawer j), and Fj (Alice ﬁnds paper in drawer j). Then the given probabilities are P(Ij) = pj and P(Fj|Ij ∩Lj) = dj. The probability of interest is P(Ij|Li ∩ ¯ Fi) for j 6= i and then for j = i. By the conditioned (on Li) version of Bayes’s rule: P(Ij|¯ Fi ∩Li) = P(¯ Fi|Ij ∩Li)P(Ij|Li)P k P(¯ Fi|Ik ∩Li)P(Ik|Li). (2) 2. Consider two dependent events A,B. Suppose P(A) = p and P(B|A) = q and P(B|Ac) = r. Find an expression in terms of p,q,r for P(A∩B|A∪B). (3) Give a clear derivation of the expression with each step labeled (e.g., deﬁnition of conditional expectation, law of total probability, etc.). 3. Using a biased coin to make an unbiased decision. Alice and Bob want to choose between the opera and the movies by tossing a fair coin. Unfortunately, the only available coin is biased (though the bias is not known exactly). How can they use the biased coin to make a decision so that either option (opera or the movies) is equally likely to be chosen? Hint: map the four outcomes of two coin ﬂips to three decisions: opera, movies, and repeat. 4. De M´er´e’s puzzle. A six-sided die is rolled three times independently. Which is more likely: a sum of 11 or a sum of 12? (This question was posed by the French nobleman de M´er´e to his friend Pascal in the 17th century.) 5. Memory. The game of memory consists of n pairs of cards where the identical cards in pair i each have the number i on them, for i = 1,...,n. The game is played by repeatedly picking two cards, and if the two cards are a pair then they are removed from the game. We will consider the case of perfect memory where if the two cards picked are not a pair they are returned to their places face up. Assume a player uses the following strategy: i) the player picks a pair if a pair is known to her, ii) else the player picks an as of yet untried card — if this card has as its pair a previously picked card the player picks the pair, else the player picks a second as of yet untried card. The game ends once all pairs have been picked. Let t(n) be the (random) number of turns until the game ends. For each n ∈ N = {5,10,15,20,25,30,35,40,45,50} run k = 1000 random realizations of the game. Thus for each n you should generate t1(n),...,tk(n) random realizations of the game. Plot ¯ t(n) = 1 kPk i=1 ti(n) versus n

Recommended supplemental problems from text. These problems are not to be handed in with your assignment. You are encouraged to work with your classmates on the supplemental problems. • §1.4. Total probability theorem and Bayes’ rule: Problems 20, 21, 22, 23, 27. • §1.5. Independence: Problems 40, 41. • §1.6. Counting: Problems 51, 53, 59. Required problems for homework: 1. Alice searches for her term paper in her ﬁling cabinet which has several drawers. She knows that she left her term paper in drawer j with probability pj 0. The drawers are so messy that even if she correctly guesses that the term paper is in drawer i, the probability that she ﬁnds it is only di. Alice searches in a particular drawer, say drawer i, but the search is unsuccessful. Conditioned on this event, show that the probability that her paper is in drawer j is given by pj 1−pidi, j 6= i, pi(1−di) 1−pidi , j = i. (1) Hint: deﬁne events Ij (term paper is in drawer j), Lj (Alice looks in drawer j), and Fj (Alice ﬁnds paper in drawer j). Then the given probabilities are P(Ij) = pj and P(Fj|Ij ∩Lj) = dj. The probability of interest is P(Ij|Li ∩ ¯ Fi) for j 6= i and then for j = i. By the conditioned (on Li) version of Bayes’s rule: P(Ij|¯ Fi ∩Li) = P(¯ Fi|Ij ∩Li)P(Ij|Li)P k P(¯ Fi|Ik ∩Li)P(Ik|Li). (2) 2. Consider two dependent events A,B. Suppose P(A) = p and P(B|A) = q and P(B|Ac) = r. Find an expression in terms of p,q,r for P(A∩B|A∪B). (3) Give a clear derivation of the expression with each step labeled (e.g., deﬁnition of conditional expectation, law of total probability, etc.). 3. Using a biased coin to make an unbiased decision. Alice and Bob want to choose between the opera and the movies by tossing a fair coin. Unfortunately, the only available coin is biased (though the bias is not known exactly). How can they use the biased coin to make a decision so that either option (opera or the movies) is equally likely to be chosen? Hint: map the four outcomes of two coin ﬂips to three decisions: opera, movies, and repeat. 4. De M´er´e’s puzzle. A six-sided die is rolled three times independently. Which is more likely: a sum of 11 or a sum of 12? (This question was posed by the French nobleman de M´er´e to his friend Pascal in the 17th century.) 5. Memory. The game of memory consists of n pairs of cards where the identical cards in pair i each have the number i on them, for i = 1,...,n. The game is played by repeatedly picking two cards, and if the two cards are a pair then they are removed from the game. We will consider the case of perfect memory where if the two cards picked are not a pair they are returned to their places face up. Assume a player uses the following strategy: i) the player picks a pair if a pair is known to her, ii) else the player picks an as of yet untried card — if this card has as its pair a previously picked card the player picks the pair, else the player picks a second as of yet untried card. The game ends once all pairs have been picked. Let t(n) be the (random) number of turns until the game ends. For each n ∈ N = {5,10,15,20,25,30,35,40,45,50} run k = 1000 random realizations of the game. Thus for each n you should generate t1(n),...,tk(n) random realizations of the game. Plot ¯ t(n) = 1 kPk i=1 ti(n) versus n

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