Engineering Probability 2018 Exam 1 - Mon 2018-02-26

Name, RCSID:

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Rules:

1. You have 80 minutes.
2. You may bring one 2-sided 8.5"x11" paper with notes.
3. You may bring a calculator.
4. You may not share material with each other during the exam.
5. No collaboration or communication (except with the staff) is allowed.
6. Check that your copy of this test has all seven pages.
7. Do any 14 of the 17 questions or subquestions. Cross out the 3 that you don't do.
8. When answering a question, don't just state your answer, prove it.

Questions:

1. (5 pts) Ten people run a race for gold, silver, bronze. How many ways can the medals be won, w/o any ties?

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2. Two teams, the Albanians and the Bostonians, are playing a 7 game series. The first team to win 4 games wins the series, and no more games are played. In any game, the Albanians have a 60% chance of winning. The games are independent, and there are no ties.

   1. (5 pts) What's the probability that the series will run to 7 games?

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   2. (5 pts) What's the probability that the Albanians win the series?

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3. (5 pts) Imagine two coins. Coin A has two heads. Coin B has the usual one head and one tail, and it is fair. You pick a coin at random (p=.5 to pick either coin) and toss it. It comes up heads.

   What is the probability that you picked coin A?

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4. (5 pts) Consider S={1,2,3,...22}. Is the set of even numbers independent of the set of multiples of 4?

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5. You are sitting an online multiple choice exam in thraumaturgy, about which you know nothing. Each question has 5 possible answers. You answer each question randomly. The exam ends when you get your first correct answer.

   1. (5 pts) What's the relevant probability distribution?

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   2. (5 pts) What's the expected number of questions you will need to answer?

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   3. (5 pts) What's the standard deviation?

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6. You are designing a check bit system for transmitting 8-bit bytes over a noisy channel. Since it's really noisy, you append two check bits to each byte, transmitting ten bits in total for each byte. Each bit, independently, can be wrong with probability \(10^{-6}\).

   You're using a Reed-Solomon error correction scheme, which we teach in another class. If there is only one bad bit in the ten transmitted bits, it will correct the byte. If there are two bad bits, it will report the error, but can't correct it. Three or more bad bits are unlikely enough that we assume they never occur.

   1. (5 pts) What's the probability that the receiver can deduce the correct byte?

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   2. (5 pts) What's the probability that the receiver will receive a byte that it knows is bad, but can't correct?

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7. Pretend that we divide the 86 field into a grid of 100 by 100 squares. 5000 students toss 10 paper airplanes each off the JEC roof. Each paper airplane has an independent and uniform probability of hitting each square. Each airplane falls into exactly one square.

   1. (5 pts) What's the mean number of airplanes to hit a particular square?

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   2. (5 pts) What's the exact probability that a particular square gets zero airplanes? It's ok to give an expression; you don't need to evaluate it.

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   3. (5 pts) What's a faster very good approximate formula? An expression is ok.

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   4. (5 pts) What's the very good approximate standard deviation for the number of airplanes to hit a particular square? An expression is ok.

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8. Chip quality control:

   1. Each chip is either good or bad.
   2. P[good]= 0.9.
   3. If the chip is good: P[still alive at t] = \(2^{-t}\)
   4. If the chip is bad: P[still alive at t] = \(3^{-t}\)

   Questions:

   1. (5 pts) What's the probability that a random chip is still alive at t=2? Give an expression and evaluate it to give a number.

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   2. (5 pts) If a random chip is still alive at t=2, what's the probability that it's a good chip?

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   3. (5 pts) If a random chip is still alive at t=2, what's the probability that it will still be alive at t=3?

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End of exam 1, total 70 points.