Suppose a box contains tickets, each labeled by an integer. Let X,Y , and Z be the results of draws at random with replacement from the box. Show that, no matter what the distribution of numbers in the box, a) P(X + Y is even) ≥1/2; b) P(X + Y + Z is a multiple of 3) ≥1/4.

Suppose a box contains tickets, each labeled by an integer. Let X,Y , and Z be the results of draws at random with replacement from the box. Show that, no matter what the distribution of numbers in the box, a) P(X + Y is even) ≥1/2; b) P(X + Y + Z is a multiple of 3) ≥1/4.

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

Suppose a box contains tickets, each labeled by an integer. Let X,Y , and Z be the results of draws at random with replacement from the box. Show that, no matter what the distribution of numbers in the box,
a) P(X + Y is even) ≥1/2;
b) P(X + Y + Z is a multiple of 3) ≥1/4.

Expert Solution

(a) We can obtain an even number if both ticket numbers are even or both are odd
Let

$x =$ probability of even number of tickets

$y =$ Probability of odd number of tickles

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