9780123869814 CHAPTER 2 Exercises Section 2.1: Experiments, Sample Spaces, and E

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9780123869814

CHAPTER 2 Exercises Section 2.1: Experiments, Sample Spaces, and Evernts An experiment consists of rolling n (six-sided) dice and recording the sum of the n ro How many outcomes are there to this experiment? 2.1 lls 2.2 (a) An experiment consists of rolling a die and flipping a coin. If the coin flip is heads, the value of the die is multiplied by -1, otherwise it is left as is. What are the possible outcomes of this experiment? Now, suppose we want to repeat the experiment in part (a) n times and reco sum of the results of each experiment. How many outcomes are there in this experiment and what are they? (b) rd the An experiment consists of selecting a number x from the interval [O, from the interval [0, 2) to form a point (x, y). (a) Describe the sample space of all points. (b) What fraction of the points in the space satisfy x>? (c) What fraction of points in the sample space satisfy x = y ? 2.3 1) and a number y 2.4 An experiment consists of selecting a point (x, y) from the interior of the unit circle. x2 + y2 (c) What fraction of the points in the space satisfy x + y> (d) What fraction of the points in the space satisfy x + y = ? Probability and Random Processes. © 2012 by Elsevier Inc. All rights reserved. 43

Explanation / Answer

Question 2.1:

Clearly as we are rolling n dice and each dice could have any number from 1 to 6, therefore there could be 6n outcomes in the throw of n dice. Now the minimum sum of the n dice would be n in case every dice has 1 on it. Also the maximum sum would be 6n in case every dice gets a six. Therefore the number of distinct sums possible for n dice throws would be 6n - (n-1) = 5n + 1

Therefore there are (5n+1) distinct sums possible from n to 6n ( inclusive )

Question 2.2:

a) Here we can have outcome of the dice as any number from 1 to 6. Now as the dice has to be multiplied in case the coin flip is heads , therefore the total number of distinct outcomes would be from 1 to 6, and also from -1 to -6

Therefore 12 distinct outcomes are there.

b) Now as we are doing this experiment n times, the minimum sum possible here would be n times the minimum value that the outcome can have that is -6. Therefore the minimum value possible here would be -6n and clearly the maximum value possible would be 6n

Therefore the total number of outcomes possible here would be 12n + 1 from -6n to 6n.

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