Final Exam, MA 3210, Introduction to Probability and Statistics, spring 2000
In the upper left corner of each page of your blue book,
mark clearly the problem number that page contains.
Useful Formula

1.
(a)
P(A|B)=0.5, P(B)=0.7. Find .
(b)
For a random variable . Find P(X >10).
(c)
What does it mean that two events A, B are independent?
(d)
How many permutations are possible to arrange the letters in "peer"?
2.
The probability that OW students like a dog is 50% and the probability that OW students like a cat is , and the probability that OW students like both a dog and a cat .
(a)
Find the probability that a randomly chosen OW student like a cat or a dog.
(b)
Find the probability that a randomly chosen OW student like both cat and dog.
3.
In a poker hand consisting of 5 cards, find the probability of holding
(a)
3 aces and 2 King.
(b)
2 spades and 2 club and one heart.
4.
Judy tosses a coin which has . If it is Heads, she draws a ball from Bag 1. If it is Tails, she draws a ball from bag 2. Bag I contains 2 red balls and 1 blue balls, Bag II contains 1 red balls and 2 blue balls. R denotes the event of drawing a red ball, B denotes the event of drawing a blue ball and H and T denote the event of getting Head and Tail respectively.
(a)
Complete the probability tree for this experiment.
(b)
Find the probability P(R|H).
(c)
(d)
Find P(R).
(e)
She drew a red ball. Find the probability that she got Heads.
5.
Long Island Railroad has 2 lines. L1, L2 which make up 60%, and 40% of LIRR service. The data shows that 3 %, 2% of each line are running late.
(a)
If a train is randomly selected, what is the probability that it is running late?
(b)
A train is randomly selected from the ones which run late. What is the probability that it is on line L2?
6.
It is known that of the cars pass the inspection. Let X be the number of the cars passing inspection. Among the randomly selected 3 cars,
(a)
Find the probability density function f(x) of X.
(b)
Find the probability that at least 2 of them pass the inspection.
(c)
Find the expected number of cars which will pass the inspection.
7.
The probability density function f(x)of a random variable X is given as

(a)
Find P(-1< X < 1).
(b)
Find the expected value of X.
8.
Let X denote the number of emergency patients admitted to the hospital : 1,2, or 3 times on any given day. Let Y denote the number of times a doctor was called on an emergency call.
 f(x,y) x 1 2 3 y 1 0.4 0. 25 0.15 2 0 0.05 0.15

(a)
Find the probability that .
(b)
Given that a doctor was called on once, find the probability that two patients were admitted.
(c)
Verify that f(x,y) is a joint probability distribution function.
9.
Given the joint density function of a random variable X and Y,

(a)
Find P( 1<X<2, Y>3).
(b)
Find the marginal distribution of X, Y.
(c)
Test if X and Y are statistically independent.
10.
A random variable X has a mean and the variance . Using Chebyshev's theorem, find P( 14< X < 26).
11.
Write down the probability distribution density function of the following.
(a)
A Poisson distribution whose expected number of arrivals per unit time is 5.
(b)
discrete uniform distribution over 5 numbers
(c)
Normal distribution with .
(d)
Standard Normal Distribution.
12.
In this problem, use the statistical tables. In each problem, indicate which distribution table you are using (Binomial, Poisson, or Normal).
(a)
A coin is biased so that P(H)=0.4. Tom tosses this coin 10 times. What is the probability of having 4 or more Heads but less than or equal to 6 Heads?
(b)
Given a normal distribution with , , Find P(21<X<27).
(c)
It is known that one out of 100 patients is allergic to grass. What is the probability that a randomly chosen 1000 patients will yield fewer than 9 patients who are allergic to grass?
(d)
A process yields 20 % defective items. If 100 items are randomly selected from the process, what is the probability that the number of defective items exceeds 16 but less than 24?