Engineering Probability Class 12 and Exam 1 - Thu 2019-02-21
Name, RCSID:
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Rules:
- You have 80 minutes.
- You may bring one 2-sided 8.5"x11" paper with notes.
- You may bring a calculator.
- You may not share material with each other during the exam.
- No collaboration or communication (except with the staff) is allowed.
- Check that your copy of this test has all six pages.
- Do any 14 of the 17 questions or subquestions. Cross out the 3 that you don't do.
- When answering a question, don't just state your answer, prove it.
Questions:
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You are trying to pass your driving test to get a driving license. You can take the test only once a month. For each time you try, you pass with probability 1/3. The random variable is the number of months until you pass for the first time.
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(5 pts) What's the relevant probability distribution?
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(5 pts) What's the expected number of months until you pass?
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(5 pts) What's the standard deviation?
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In this year 2019, which has 365 days, the set of outcomes is the set of days of the year, from Jan 1 to Dec 31. Event A is that the day is Monday. Event B is that the day is in Jan. Here is a calendar for Jan:
January 2019 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
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(5 pts) What is the probability of B?
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(5 pts) Are A and B independent? Prove your answer (don't just state it).
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You are scanning a B&W page and transmitting it over a noisy channel. A black bit is coded as 1 and a white bit as 0. Event A is that a random bit is black. P(A) = .01. Event B is that the bit is received as black. Sometimes the bit is changed during transmission. 1% of the white bits are changed to black. But 10% of the black bits are changed to white.
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(5 pts) What is the probability of B?
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(5 pts) What is P(A and B))?
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(5 pts) What is P(A' and B')?
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(5 pts) What is the probability that the bit arrived correct?
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(5 pts) What is P(A|B), the probability that 1 was transmitted, if you received 1?
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(5 pts) What is P(A' | B'), the probability that 0 was transmitted, if you received 0?
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(5 pts) Now you take that received page and transmit it a second time over the same noisy channel. Let event C be that you receive a black bit the 2nd time. Compute P(C).
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(5 pts) An LCD display has 1000 * 1000 pixels. A display is accepted if it has 15 or fewer faulty pixels. The probability that a pixel is faulty coming out of the production line is 1e-5. Find the proportion of displays that are accepted.
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Pretend that there are only 28 students in this class. This is about their birthdays. Assume that they are uniformly distributed over the year (although this is actually false.)
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(5 pts) What's the probability that no one has a birthday on Feb 29? (Year 2000 was a leap year.) As always a reasonable formula is ok.
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(5 pts) Now pretend that no one has a birthday on Feb 29. What's the probability everyone has a different birthday?
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(5 pts) Assume that RPI has 7000 students, and their birthdays are uniformly distributed and no one is on Feb 29. Use a reasonable approximate distribution to compute the probability that exactly 20 students' birthday is today, Feb 21.
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(5 pts) Able and Baker take turns tossing a coin until one gets Head. The winner is the person who made that toss. So, if Able gets a head on the first toss, he wins. If Able tosses Tail and then Baker tosses Head, Baker wins. And so on. What's the probability that Able eventually wins the game?
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End of exam 1, total 70 points.