Exam 1 Classical Probability Example Problems

Suppose the committee decides to randomly choose seven graduate students from a pool of 30 applicants, of whom 20 are foreign and 10 are US applicants. What is the probability that a chosen seven will have four foreign students and three US students?

(20C4x10C3)/(30C7) = .2856

Suppose we are given a population with the following distribution: P(AA) = p, P(Aa) = 2q, P(aa) = r. Alleles are randomly donated form parents to offspring. Assuming random mating, what is the probability that the mating is Aa x Aa and the offspring is aa?

4q^2x(1/4) = q^2

In how many different ways can a student club with 500 members choose its president and vice president?

500x499 = 29500 ways

40% of students were in the top 10% of their high school class, and 65%, of whom 25% were in the top 10% of their class, are white. What is the probability that a student selected randomly was either in the top 10% of their class or is white?

80%

Suppose that three types of anti-missile defense systems are being tested. The first will detect the target 10 out of 12 times, the second will detect the target 9 out of 12 times, and the third will detect the target 8 out of 12 times. We have observed that a target has been detected. What is the probability that the defense system was the third type?

Baye's rule ((1/3)x(8/12))/((1/3)x(10/12)+(1/3)x(9/12)+(1/3)x(8/12)) = 8/27

We toss two balanced dice, and let A be the event that the sum of the face values of two dice is 8, and B be the event that the face value of the first is 3. Calculate P(A|B).

Conditional probability (1/36)/(1/6) = 1/6

A fruit basket contains 25 apples and oranges, of which 20 are apples. If two fruits are randomly picked in sequence, what is the probability that both the fruits are apples?

Conditional probability (20/25)x(19/24) = .633

In a tank containing 10 fish, there are three yellow and seven black fish. We select three fish at random. (a) What is the probability that exactly one yellow fish gets selected? (b) What is the probability that at most one yellow fish gets selected? (c) What is the probability that at least one yellow fish gets selected?

(a) (3C1x7C2)/(10C3) = .525 (b) (3C1x7C2)/(10C3) + (3C0x7C3)/(10C3) = .525 + .292 = .817 (c) 1 - (3C0x7C3)/(10C3) = 1 - .292 = .708

A die is loaded such that the probability that the number shows up is Ki, i = 1, 2, ..., 6, where K is a constant. (a) Find the value of K. (b) Find the probability that a number greater than 3 shows up.

(a) 1/21 (b) 15/21

A subway station has 12 gates, six inbound and six outbound. The number of gates open in each direction is observed at a particular time of day. (a) Define a suitable sample space. (b) What is the probability that at most one gate is open in each direction? (c) What is the probability that at least one gate is open in each direction? (d) What is the probability that the number of gates open is the same in both directions? (e) What is the probability of the event that the total number of gates open is 6?

(a) S={(0,0)(1,0)(2,0)...(4,6)(5,6)(6,60} 49 outcomes (b) 4/49 (c) 36/49 (d) 1/7 (e) 1/7

In a room there are n people. What is the probability that at least two of them have a common birthday?

1 - (365x364x...x(365-n+1))/365^n

Suppose a statistics class contains 70% male and 30% female students. It is known that in a test, 5% of males and 10% of females got an A. If one student is randomly selected and observed to have an A grade, what is the probability that it is a male student?

Law of total probability (.7x.05)/(.3x.1+.7x.05) = 7/13

During an epidemic in a town, 40% of its inhabitants became sick. Of any 100 sick persons, 10 will need to be admitted to an emergency ward. What is the probability that a randomly selected person from this town will need to be admitted to an emergency ward?

Law of total probability .4x.1 + .6x0 = .04

Assume that a noisy channel independently transmits symbols, "0" 60% of the time and "1" 40% of the time. At the receiver, there is a 1% chance of obtaining any particular symbol distorted. What is the probability of receiving a 1?

Law of total probability .6x.01 + .4x.99 = .402

Suppose that we toss two fair dice. Let E1 denote the event that the sum of the dice is 6 and E2 denote the event that the first die equals 4. Are E1 and E2 independent events?

P(E1∩E2) = 1/36 =/ P(E1)P(E2) = 5/216 No

An urn contains 15 balls numbered 1-15. If four balls are drawn at random, with replacement and without regard for order, how many samples are possible?

Sampling with replacement and the objects are not ordered (15+4-1)C4 = 18C4 = 3060

How many different ways can the admissions committee choose four foreign students from 20 foreign applicants and 3 students from 10 US applicants?

Sampling without replacement and the objects are not ordered 20C4x10C3 = 581400 ways

How many distinct three-digit numbers can be formed using the digits 2, 4, 6, and 8 if no digit can be repeated?

Sampling without replacement and the objects are ordered 4P3 = 4!/(4-1)! = 24