Midterm_& homework

Midterm_& homework - Select a person at random at...

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Unformatted text preview: Select a person at random at a local baseball game. Consider these events: C = person buys soda, B = person buys beer, N = person buys peanuts Consider these probabilities: a) The probability the person buys none of the things. b) The probability the person buys exactly one of the things c) The probability the person buys all three, given that he buys at least 2. d) The probability the person buys peanut, given that he doesn’t buy beer 1. Calculate (a) ­(d) assuming that P (C ) = .40 , P ( B) = .60 , P ( N ) = .30 , P (C ∩ B ∩ N ) = .08 , P (C ∩ B) = .17 , P (C ∩ N ) = .20 , P ( B ∩ N ) = .15 € € 2. Calculate (a) ­(d) assuming that P€ ) = .40 , P€ ) = .60 , P ( N ) = .30 and that C, B, N (C (B € € € are independent events. € € € 3. A pair of 6 ­sided die is tossed. If the sum of the two die is at least 10, you win $10. If the sum of the die is a number between 4 and 9, you win $5. If the sum of the die is a number less than 4, you win $0. a) What are the possible winnings? b) What is the probability for each possibility, assuming that the die are fair? c) What is the expected winnings? d) What is the variance of winnings? 4. Let X be the payout from the previous problem. Compute a) E (1 / X ) , b) E ( X ) c) E (2.3 X − 1.5) € € 5. Puenaa is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, € it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Puenaa's wedding? 6. A randomly selected vehicle has a 10% chance of failing inspection. A simple random sample of 12 vehicles is selected. a) What is the mean and standard deviation number of vehicles in the sample that fail? b) What is the probability that at least 1 vehicle fails? c) What is the probability that at least 2 vehicles fail? 6. (continuation of 6) Suppose vehicles are inspected one by one without stopping. a) What is the probability that the first vehicle to fail is the 6th one inspected? b) What is the probability the second vehicle to fail is the 8th one inspected? 7. Cars arrive at a gas station in a remote part of the desert according to a Poisson process. On average, 0.30 cars arrive per hour. a) What is the probability that no cars arrive between 12:00noon and 12:30pm? b) What is the probability that 4 or fewer cars arrive between noon and 8pm? 8. On a quiz, the mean quiz score was 7.7 and the median is 7. The first quartile is 3.5 and the third quartile is 9. The maximum and minimum scores are 10 and 0 respectively. The standard deviation of scores is 1.5. There are 40 students in the class a) Suppose four students with zeros had their scores changed to 1. Does this effect the mean? How? Does this effect the median? How? c) Suppose now that instead of what was done in part (a), every student had one point added to their score. How does this effect the mean? The median? The standard deviation? 9. Imagine that you encounter three traffic lights in a row as you pass through three intersections. The event GGG means that you did not have to stop at any of the lights. The event GSG means that you had to stop only at the second light. a) list all 8 outcomes of this sample space b) list the outcomes for the event of stopping at exactly one light c) If the lights operate independently, and you have a 0.3 probability of stopping at any one of the lights, find the probability that, in passing though the three intersections, you have to stop at exactly one light d) Suppose now the lights are not independent. Now assume that if you are stopped at one light, the probability you have to stop at the next light is 0.1, and that if you pass through one light, the probability you have to stop at the next light is 0.4. Assume also that the first light you encounter has a .30 probability of stopping you. Compute the probability that, in passing the through the three intersections, you stop at exactly one light. ...
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This note was uploaded on 10/16/2011 for the course MATH 324 taught by Professor Staff during the Spring '08 term at S.F. State.

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