Problems in Probability

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Fall 2011 Strategic Practice 1: Section 1 (Naive Definition of Probability) - Question 1
For each part, decide whether the blank should be filled in with =, <, or >, and give a short but clear explanation.
(a) (probability that the total after rolling 4 fair dice is 21) __ probability that the total after rolling 4 fair dice is 22)
(b) (probability that a random 2 letter word is a palindrome) __ (probability that a random 3 letter word is a palindrome)
Solution: (a) All ordered outcomes are equally likely here. So for example with two dice, obtaining a total of 9 is more likely than obtaining a total of 10 since there are two ways to get a 5 and a 4, and only one way to get two 5s. To get a 21, the outcome must be a permutation of (6, 6, 6, 3) (4 possibilities), (6, 5, 5, 5) (4 possibilities), or (6, 6, 5, 4) (4!/2 = 12 possibilities). To get a 22, the outcome must be a permutation of (6, 6, 6, 4) (4 possibilities), or (6, 6, 5, 5) (4!/22 = 6 possibilities). So getting a 21 is more likely; in fact, it is exactly twice as likely as getting a 22. (b) The probabilities are equal, since for both 2-letter and 3-letter words, being a palindrome means that the first and last letter are the same.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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