Exercise+Sets


 * CS 197 AIM**
 * Exercise Set 1 (Second Sem, AY2011)**
 * Deadline: 2:30 pm, Nov 19, 2011**

1. Suppose a fair coin is tossed repeatedly until both a head and tail have appeared at least once.

(a) Describe the sample space Ω

(b) What is the probability that three tosses will be required ?

2. There are three cards. The first is green on both sides, the second is red on both sides and the third is green on one side and red on the other. We choose a card at random and we see one side (also chosen randomly). If the side we see is green, what is the probability that the other side is also green ?

3. The probability that a child has blue eyes is 1/4. Assume independence between children. Consider a family with 3 children.

(a) If it is known that at least one child has blue eyes, what is the probability that at least two children have blue eyes ?

(b) If it is known that the youngest child has blue eyes, what is the probability that at least two children have blue eyes ?

4. Suppose that 30% of computer owners use a Mac, 50% use Windows, and 20% use Linux. Suppose that 65% of the Mac users have been infected by a computer virus, 82% of Windows users get the virus, and 50% of the Linux users get the virus. We select a person at random and learn that his system was infected with the virus. What is the probability that he is a Windows user ?

5. A box contains 5 coins and each has a different probability of showing heads. Let p1, ..., p5 denote the probability of heads on each coin.

Suppose that

p1 = 0, p2 = 1/4, p3 =1/2 , p4 = 3/4 , p5 = 1

Let H denote ”heads is obtained” and let Ci denote the event that coin i is selected.

(a) Select a coin at random and toss it. Suppose a head is obtained. What is the posterior probability that coin i was selected (i = 1, ..., 5) ?

(b) Toss the coin again. What is the probability of another head ?

(c) Now suppose that the experiment was carried out as follows. We select a coin at random and toss it until a head is obtained. Find P (Ci |B4 ) where B4 = ”first head is obtained on toss 4”.