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(solution) All questions are about probability, but i want details of each

All questions are about probability, but i want details of each step.

Math 170A - Homework 1 - Due 4 October 2016

1. Let A = {1, 3, 4, 5}, B = {2, 4, 6, 7} and C = {1, 4}. Find (A ? B) ? C c .

2. Suppose that a 1 meter glass rod falls onto the floor and is broken into two pieces. Assume

that the breaking point is uniform between 0 and 1 meter. What is the probability that

the longer piece has length at least 0.6 meter?

3. Suppose that A and B are two events such that P(A) = 0.4, P(B) = 0.5 and P(A?B) = 0.6.

Find P(A ? B).

4. Let A, B, C be sets. Show that

(i) A ? B c = A ? (A ? B)c and

(ii) A ? C c ? (A ? B c ) ? (B ? C c ).

5. If A1 and A2 are two sets, we define the symmetric difference A1 ?A2 by

A1 ?A2 = (A1 ? Ac2 ) ? (A2 ? Ac1 ).

(This is simply the set of all points that either in A or B but not both.) Show that

(i) P(A1 ?A2 ) = P(A1 ) + P(A2 ) ? 2P(A1 ? A2 ) and

(ii) for every sets A, B, C,

P(A?C) ? P(A?B) + P(B?C). 6. Suppose that I arrive at a bus stop at some time between 8:00 am and 8:25 am with

uniform probability. Buses arrive at 8:03, 8:13, 8:23, 8:33,. . .. What is the probability

that I need to wait more than 2 minutes? 1

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