Answered: probability distribution of…

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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Question

An urn contains 1 red ball, 2 black balls and 3 white balls. Each time select a ball from the urn with replacement. Now if we select twice and let X,Y,Z represent the number of times that red ball, black ball or white ball is select, respectively. Calculate P(X=1|Z=0), Find the probability distribution of two-dimensional random variable (X,Y)

Expert Solution

Step 1: in this step, we have discussed the probabilities

To calculate $P (X = 1 | Z = 0)$ , we need to find the conditional probability that the red ball is selected once ( $X = 1$ ) given that the white ball is not selected in the two draws ( $Z = 0$ ).

Let us break down the problem step by step:

The probability of selecting the red ball on any single draw (with replacement) is $1 / 6$ , as there is 1 red ball out of 6 total balls.
The probability of not selecting the white ball on any single draw is $4 / 6$ , as there are 4 non-white balls out of 6 total balls.
Since you're drawing twice with replacement, the probability of not selecting the white ball in both draws is $(4 / 6) * (4 / 6) = (2 / 3)^{2} .$