Example 2.3.4 (Polya's urn problem) From an urn containing r red and sblack balls, n (nsr + s) balls are drawn successively, such that the balldrawn is always replaced and, in addition, q balls of the same colour areadded to the urn. To find the probability of a complete run of n black balls,let A denote the event that the ith ball is black (i = 1, 2, . . . , n). The requiredevent is A1N A2n. n Ap.%3D

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Answered: Example 2.3.4 (Polya's urn problem)…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Fill in the gaps with one or more words: A.B,M and X are random variables. If A and B are independant then they _____ uncorrelated. If E(MX)= E(M)E(X) then M and X _____ independant.
Suppose 21% of trees in a forest have severe leaf damage from air pollution. If 6 trees are selected atrandom,(a) find the probability that:i. exactly four of the selected trees have severe leaf damage; ii. no more than two of the selected have severe leaf damage;iii. exactly five of the selected trees do not have severe leaf damage;iv. at least five of the selected trees have severe leaf damage;v. less than two of the selected trees have severe leaf damage.(b) find the expected number of trees with severe leaf damage.
The resumes of two male applicants for a college teaching position in Engineering Data Analysis are placed in the same file as the resumes of two female applicants. Two positions become available, and the first, at the rank of assistant professor, is filled by selecting one of the foUr applicants at random. The second position, at the rank of instructor, is then filled by selecting at random one of the remaining three applicants. Using the notation M,F1, for example, to denote the simple event that the first position is filled by the second male applicant and the second position is then filled by the first female applicant. Find the probability that the position of assistant professor is filled by a male applicant. O 1/3 O 1/6 O 2/3 O 1/2
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If a variable can take certain integer values between two given points, then it is called O a. Continuous random variable O b. Probabilistic random variable c. O c. Deterministic random variable O d. Uncertain random variable O e. Discrete random variable
3. An unloaded, 4-sided dice is rolled twice. Let X be the maximum number obtained from the two rolls, and Y be the minimum number from the two rolls. (The dice can show the numbers 1, 2, 3, or 4, each with probability 1/4. As an example, if the two rolls are 1 followed by 3, then X = 3 and Y = 1.) (a) Find the joint pmf of X and Y. (b) Find the conditional pmf of X given that Y = 2. (Hint: part (a) is similar to a problem from Modelling Uncertainty. Please express your answers in table form, and please use fractions in lowest terms.)
Let S = {E1, E2, E3, EA} be the sample space of an experiment and let = {E1, E2}. B = {E2, E3}, and C = of the sample points are assigned as follows: P(E) P(E3) = 0.30, and P(E4) = 0.40. Find P(AU B). {Es, E,} be events from S. The probabilities 0.10. P(E2) = 0.20, A = %3D Select one: O a. 0.60 b. 0.20 O c. 0.90 O d. 0.30 e. 0.70
n = {1, 2, 3, 4} with each point carrying probability 1/4. Let A1 = {1, 2}, A2 = {1, 3}, A3 = {1, 4}. Then any two of A1, A2, A3 are independent, but A 1 , A 2, A 3 are not independen
Example 15: Three bags contain 3 red, 7 black; 8 red, 2 black, and 4 red & 6 black balls respectively. 1 of the bags is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the third bag.
Example 8.15 Suppose we have an urn containing ten balls of different colours such that 2 balls are red and dotted 1 ball is green and dotted 4 balls are red and striped 3 balls are green and striped What is the probability of drawing any particular ball from this urn?
2. Page 34, 1.4.14*. Each of three persons fires one shot at a target. Let Ck denote the event that the target is hit by person k, k = 1, 2, 3. If C1, C2, C3, are independent and if P(C1) = P(C2) = 0.8, and P(C3) = 0.9, compute the probability that (a) all of them hit the target; (b) exactly one hits the target; (c) no one hits the target; (d) at least one hits the target.

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