The first assignment of Stochastic Process.
1. Two dice are rolled. What is the probability that at least one is a six? If the two faces are different, what is the probability that at least one is a six?
2. Suppose that 5 percent of men and 0.25 percent of women are color-blind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females.
3. A and B play until one has 2 more points than the other. Assuming that each point is independently won by A with probability p, what is the probability they will play a total of 2n points? What is the probability that A will win?
4. Two cards are randomly selected from a deck of 52 playing cards.
(a) What is the probability they constitute a pair (that is, that they are of the same denomination)?
(b) What is the conditional probability they constitute a pair given that they are of different suits?
5. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define eventsE1, E2, E3, and E4 as follows:
E1={the first pile has exactly 1 ace}, E2={the second pile has exactly 1 ace}, E3={the third pile has exactly 1 ace}, E4={the fourth pile has exactly 1 ace}
Find P(E1,E2,E3,E4).
Hint: Use P(E1,E2···,En)=P(E1)P(E2|E1)P(E3|E1E2)···P(En|E1···En−1) 6- The event A and B is mutually exclusive, can they be independent?
7-Suppose K identical boxes contain n balls numbered 1 through n. One ball is drawn from each box.
What is the probability that m is the largest number drawn?
What is the probability that K drawn balls have identical numbers?
What is the probability that K drawn balls have different numbers? Discus in different values K and n.
8- A player tosses a penny from a distance onto the surface table ruled in 1 in. squares. If the penny is ¾ in. in diameter, what is the probability that it will fall entirely inside a square (Suppose that the penny lands on the table) 9- Box 1 contains 100 bulbs of which 10% are defective. Box 2 contains 2000 bulbs of which 5% are defective. Two bulbs are picked from a randomly selected box.
A) Find the probability that both bulbs are defective?
B) Assuming that both are defective, find the probability that it came from box 1?
10- By an example prove the following equations:
= −
− + −
; = . −
− ; ∑ =
11- Suppose a die is rolled twice. What are the possible values that the following random variables can take on?
(a) The maximum value to appear in the two rolls.
(b) The minimum value to appear in the two rolls.
(c) The sum of the two rolls.
(d) The value of the first roll minus the value of the second roll.
12-A coin is tossed an infinite number of times. Show that the probability that k heads are observed at the nth toss but not earlier equals −
−
