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homewk1sols - IEOR 4106 Introduction to Operations Research...

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IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review : Read Chapters 1 and 2 in the textbook, Introduction to Proba- bility Models , by Sheldon Ross. Please do the six problems marked with an asterisk and turn them in next Tuesday. These problems are written out in case you do not yet have the textbook (available in the Columbia Bookstore). Extra problems are provided in case you need extra review. Solutions are provided for all these problems. CHAPTER 1 *Problem 1.3. A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times? ——————————————————————————- The sample space is a set with countably infinitely many elements. The sample space is the set S ≡ { ( e 1 , e 2 , . . . , e n ) : n 2 } , where e i ∈ { H, T } (i.e., e i is either heads or tails) and, in addition, e n = e n - 1 = H and, for 1 i n - 2 (for n 3), if e i = H , then necessarily e i +1 = T . Here are initial ele- ments of S : { H, H } , { T, H, H } , { T, T, H, H } , { H, T, H, H } , { T, T, T, H, H } , { T, H, T, H, H } , { H, T, T, H, H } , { T, T, T, T, H, H } , . . . ——————————————————————————- *Problem 1.21 Suppose that 5% of men and 0 . 25% of women are color-blind. A color-blind person is chosen at random. What is the probability of this person being male? (Assume that there are an equal number of males and females.) ——————————————————————————- Let C be the event that a person chosen at random is color-blind. Let M be the event that a random person is male. Let M c be the event that a random person is female, the complement of the event M . Then P ( M | C ) = P ( MC ) P ( C ) = P ( C | M ) P ( M ) P ( C | M ) P ( M ) + P ( C | M c ) P ( M c ) = 0 . 05 × 0 . 5 (0 . 05 × 0 . 5) + (0 . 0025 × 0 . 5) = 20 21 . ——————————————————————————- Problem 1.22 ———————————————————————– The calculation becomes easy if we view it in a good way. One might observe that the experiment has the form of tennis once deuce has been reached or table tennis (ping pong)
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after the score of 20 -
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