MA519_JAN06 - QUALIFYING EXAMINATION JANUARY 2006 MATH 519...

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QUALIFYING EXAMINATION JANUARY 2006 MATH 519 - Prof. Dasgupta NOTE : 1. ANSWER TWO QUESTIONS FROM EACH GROUP. IF YOU ANSWER MORE THAN TWO QUESTIONS, ONLY THE FIRST TWO YOU HAVE ANSWERED WOULD BE GRADED. THE OTHERS WILL NOT GET GRADED. 2. STANDARD NORMAL TABLE MAY BE REQUIRED FOR SOME PROBLEMS. Group A 1. The birthdays of 5 people are known to fall in exactly 3 calendar months. What is the probability that exactly two of the 5 were born in January ? 2. Coupons are drawn, independently, with replacement, one at a time, from a set of 10 coupons. Find, explicitly, the expected number of draws a) until the first drawn coupon is drawn again; b) until a duplicate occurs. 3. Let N be a positive integer. Choose an integer at random from { 1 , ..., N } . Let E be the event that your chosen random number is divis- ible by 3, and divisible by at least one of 4 and 6, but not divisible by 5. Find, explicitly, lim N →∞ P ( E ). 4. Anirban is driving his Dodge on a highway with 4 lanes each way. He is wired to change lanes every minute on the minute. He changes with equal
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