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15. We flip a fair coin three times; these flips are independent of each other. These three coin flips give us a sequence of length three, where each symbol is H or T. Define the events A- B = the sequence contains at most one T. the symbols in the sequence are not all equal Which of the following is true? (a) The events A and B are independent. (b) The events A and B are not independent (c) None of the above.

16. Let n 2 2 be the number of students who are writing this exam. Each of these students has a uniformly random birthday, which is independent of the birthdays of the other students. We ignore leap years; thus, the year has 365 days. Define the event A-at least two students have their birthday on December 14. What is Pr(A)? (a) (:) . (斋)2. (b) 1- (n 5)-2 2 (364n-2 2 365 c) 1- (364)n 1. (364) 365 365

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157 A and B are independent if there is no common case between A & B Sample space of A:{HHH,HIT, HTH, THI} Sample space of B:

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