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Department of EECS - University of California at Berkeley EECS 126 - Probability and Random Processes - Spring 2008 Midterm 1: 2/28/
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There are two coins. Coin 1 is fair. Coin 2 is such that P (H) = 0.6.
You flip the two coins together repeatedly. What is the probability that coin 1 yields H before coin 2?
You are given one of the two coins, with equal probabilities. You flip the coin twice and you get H both times. What is the probability that you got coin 1?
You throw a dart at a circular target with radius 1. You miss the target with probability 0.2. If you hit the target, the dart location is uniformly distributed inside the target. Let X be the distance from dart to the center of the target when you hit it and X = 2 when you miss the target.
What is the p.d.f. of X;
Plot the c.p.d.f. of X;
Calculate V ar(X), the variance of X.
Assume that humanity will either survive 10 billion years or ten million years, with equal prob- abilities. For simplicity, assume that the population is constant and about equal to 8 billion people, in both cases. Assume also that you are picked randomly human, among all humans who will ever live. You observe that humanity has been around for about 5 million years. What is the probability that humanity will survive ten billion years, given your observation?
A randomly picked 126 student has a 20% chance of being a genius and an 80% chance of being very smart but somehow short of genius. A genius gets a score on the first midterm that is uniformly distributed in [70, 100]. A very smart student gets a score that is uniformly distributed in [0, 100]. A genius has a probability 80% of going to graduate school and a very smart student has a probability 20% of going to graduate school. What is the probability that a randomly picked student who gets a score of 80 will go to graduate school?
