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Course: ECON 3190, Fall 2008School: Cornell
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319, Econ Fall 2008 TA: Simon Kwok Handout 4 1. The inhabitants of an island tell the truth one third of the time, and lie with probability 2/3. On an occasion, after one of them made a statement, another islander stepped forward and declared the statement true. What is the probability that the rst statement was indeed true? 2. In an o ce there are 200 employees. None of the employees are married to each other....
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319, Econ Fall 2008
TA: Simon Kwok
Handout 4
1. The inhabitants of an island tell the truth one third of the time, and lie with probability 2/3. On an occasion, after one of them made a statement, another islander stepped forward and declared the statement true. What is the probability that the rst statement was indeed true? 2. In an o ce there are 200 employees. None of the employees are married to each other. Suppose that a person is chosen at random. The probability of that person being a man and 1 being married is 10 . If you know that the person is married, the probability of the person 2 being a man is 9 . If you know that the person is single, the probability of the person being 4 a man is 11 . (a) How many married people are in the o ce? (b) How many married women are in the o ce? (c) If you know that the random person chosen is a woman, what is the probability that she is single? 3. Consider the following two functions (a) f (x) = (b) f (x) = C(2x x3 ) if 0 < x < 5 2 0 otherwise,
3 C(2x x2 ) if 0 x < < 2 0 otherwise.
Can (a) be a probability density function (pdf)? If so, determine C. Can (b) be a pdf? If so, determine C. 8 x< 2 >0 > > > 0:4 2 x<0 < 0:5 0 x<1 : 4. Suppose a random variable X has a cdf:F (x) = > > 0:8 1 x<4 > > : 1 x4 (a) Find the probability (mass) function f (x) and give your reasoning. 1. Pr(x 1); 2. Pr(x > 1 ); 2 3. Pr(x = 0): 5. (For those who are up to more challenge. Not for exam) Consider two gamblers, A and B. A has $5 and B has $10. They play a game in which they a fair coin. If it comes up a head ip then A gives B a dollar. If it comes up a tail then B gives A a dollar. They continue like this until one player has no money left other player is then considered to be the winner. the What is the probability that A wins this game? [Hint: Let p(i) denote the probability that a player wins given that this payer has $i. Use conditional probability to show the recursion (i + 1)p(i) = ip(i + 1) by induction.] (b) Find the values for:
1
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