asst123 - York University MATH 2030 3.0(Elementary...

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York University MATH 2030 3.0 (Elementary Probability) Fall 2011 Assignment 1 –Solutions, Sept 2011 § 1.1 No. 2 (a) We use an equally likely outcomes model with Ω = { suppose, a, word, is, picked, at, random, from, this, sentence } . This Ω has 10 elements. The event in question is A = { suppose, word, picked, random, from, this, sentence } which has 7 el- ements. So P ( A ) = 7 / 10. (b) Now we have an event B = { suppose, picked, random, sentence } , and P ( B ) = 4 / 10. (c) A B = B in this particular case, so P ( A B ) = P ( B ) = 4 / 10. § 1.3 No. 4 (a) Yes: the event is { 0 , 1 } (b) Yes, this is just another way of saying there was exactly one head: the event is { 1 } . (c) No, the event can’t be expressed in this model. This particular model lets us keep track of the number of heads, but not the order they occur in. (d) Yes: the event is { 1 , 2 } . § 1.3 No. 6 (a) There are 10 words, each of which is equally likely to be picked. I’ll use an equally likely outcomes model with sample space consisting of { suppose, a, word, is, picked, at, random, from, this, sentence } . Let X be the random variable that counts the length of the word, eg X (suppose) = 7, X (word) = 4, etc. For each x we have to count the number of outcomes (ie words in the sentence) of length x , and then divide by 10 to get the desired probabilities. We compute that x 1 2 3 4 5 6 7 8 P ( X = x ) 1 10 2 10 0 3 10 0 2 10 1 10 1 10 (b) We have the same model, but a different random variable Y that counts the number of vowels in a word. For example, Y (suppose) = 3, Y (sentence) = 3. Now for every y we have to count the number of words with y vowels in order to compute probabilities. Doing this we get that y 1 2 3 P ( Y = y ) 6 10 2 10 2 10 1
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2 You are free to add other possible values y to your table, but the associated probabilities will be 0. § 1.3 No. 9 One way to do this is to fill in probabilities for all regions in a Venn diagram, and then add up those probabilities as needed. This would give that P ( F G H ) = 0 . 1 P ( F G H c ) = P ( F G ) - P ( F G H ) = 0 . 4 - 0 . 1 = 0 . 3 P ( F G c H ) = P ( F H ) - P ( F G H ) = 0 . 3 - 0 . 1 = 0 . 2 P ( F c G H ) = P ( G H ) - P ( F G H ) = 0 . 2 - 0 . 1 = 0 . 1 P ( F G c H c ) = P ( F ) - P ( F G H c ) - P ( F G c H ) - P ( F G H ) = 0 . 7 - 0 . 3 - 0 . 2 - 0 . 1 = 0 . 1 P ( F c G H c ) = P ( G ) - P ( F G H c ) - P ( F c G H ) - P ( F G H ) = 0 . 6 - 0 . 3 - 0 . 1 - 0 . 1 = 0 . 1 P ( F c G c H ) = P ( H ) - P ( F G c H ) - P ( F c G H ) - P ( F G H ) = 0 . 5 - 0 . 2 - 0 . 1 - 0 . 1 = 0 . 1 (a) P ( F G ) is the sum of the six terms above, other than P ( F c G c H ) . So it equals 0.9 (b) P ( F G H ) is the sum of all the terms above, namely 1.0 (c) This is the last term above, namely 0.1 Alternatively one can use inclusion/exclusion: (a) P ( F G ) = P ( F ) + P ( G ) - P ( F G ) = 0 . 7 + 0 . 6 - 0 . 4 = 0 . 9 (b) P ( F G H ) = P ( F ) + P ( G ) + P ( H ) - P ( F G ) - P ( F H ) - P ( G H ) + P ( F G H ) = 0 . 7 + 0 . 6 + 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 + 0 . 1 = 1 . 0
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