Three-sided die. On any roll of this die, the result is 1 with probability 1/2, 2 with probability 1/4, and 3 with probability 1/4. Consider a sequence of six independent rolls of this die (a) Find the probability that exactly two of the rolls result in a 3. (b) Given that exactly two of the six rolls resulted in a 1, find the probability that the first roll resulted in a 1. (c) We are told that exactly three of the rolls resulted in a 1 and exactly three rolls resulted in a 2. Given this information, find the probability that the six rolls resulted in the sequence (1, 2, 1, 2, 1, 2). (d) Find the conditional probability that exactly k rolls resulted in a 3, given that at least one roll resulted in a 3.
Three-sided die. On any roll of this die, the result is 1 with probability 1/2, 2 with probability 1/4, and 3 with probability 1/4. Consider a sequence of six independent rolls of this die (a) Find the probability that exactly two of the rolls result in a 3. (b) Given that exactly two of the six rolls resulted in a 1, find the probability that the first roll resulted in a 1. (c) We are told that exactly three of the rolls resulted in a 1 and exactly three rolls resulted in a 2. Given this information, find the probability that the six rolls resulted in the sequence (1, 2, 1, 2, 1, 2). (d) Find the conditional probability that exactly k rolls resulted in a 3, given that at least one roll resulted in a 3.
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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Transcribed Image Text:Three-sided die. On any roll of this die, the result is 1 with probability 1/2, 2 with probability
1/4, and 3 with probability 1/4. Consider a sequence of six independent rolls of this die
(a) Find the probability that exactly two of the rolls result in a 3.
(b) Given that exactly two of the six rolls resulted in a 1, find the probability that the
first roll resulted in a 1.
(c) We are told that exactly three of the rolls resulted in a 1 and exactly three rolls
resulted in a 2. Given this information, find the probability that the six rolls resulted
in the sequence (1, 2, 1, 2, 1, 2).
(d) Find the conditional probability that exactly k rolls resulted in a 3, given that at
least one roll resulted in a 3.
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