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hw4sol

# hw4sol - Math 361 X1 Homework 4 Solutions Spring 2003...

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Math 361 X1 Homework 4 Solutions Spring 2003 Graded problems: 1(b)(d); 3; 5; 6(a); As usual, you have to solve the problems rigorously, using the methods introduced in class. An answer alone does not count. The problems in this assignment are intended as exercises in (i) modeling a given word problem within the success/failure framework (which requires, in particular, to say what a “trial” corresponds to and what “success” means, to specify the parameters n and p , and to express the events in question as success/failure events), and (ii) applying appropriate approximations to the binomial distribution (which, besides knowing the relevant formulas, requires deciding which approximation is appropriate). Therefore, you will not get credit if you attempt to do these problems by brute force, using the binomial distribution (which, in most cases, would be more difficult or impossible anyway). Problem 1. Find formulas for the following probabilities. You can leave answers in “unsimplified” form. How- ever, in those cases where approximate formulas are requested (parts (b) and (d)), these formulas should be simple enough so that one could easily compute a numerical value with the aid of a basic, non-programmable calculator. Thus, for example, a summmation involving 100 terms would not be acceptable, nor would formulas involving large factorials (such as 100! or binomial coefficients). (a) An exact formula for probability that in a class of 365 students at least two have their birthday on January 1. (You may assume that there are 365 possible birthdays.) (b) An approximate formula for the probability computed in (a). (c) An exact formula for the probability that in a country with a population of 365,000,000 people exactly 1,000,000 have their birthday on January 1. (d) An approximate formula for the probability computed in (c). Solution. (a) Interpreting the 365 students as 365 success/failure trials with success meaning that the stu- dent’s birthday falls on January 1 (which occurs with probability p = 1 / 365), the probability in question is P ( 2 successes) = 1 - P (0) - P (1) = 1 - 1 - 1 365 365 - 365 1 1 365 1 1 - 1 365 364 (= 0 . 2642408 . . . ) Remark: Despite the fact that the problem involves birthdays, this problem is mathematically quite different from the birthday problem and must be treated within a success/failure model. The difference is that the issue here is whether or not someone’s birthday falls on a specific date , not whether two people have the same birthday. (b) Since p = 1 / 365 is small and equal to 1 /n , Poisson approximation should be used. With μ = np = 1, we get for the probability in (a) the approximate value 1 - P (0) - P (1) = 1 - e - μ μ 0 0! - e - μ μ 1 1! = 1 - (1 + μ ) e - μ = 1 - 2 e - 1 (= 0 . 2642411 . . . ) Remark: Poisson approximation is ideally suited for this case since p is exactly 1 /n and the probabilities P ( k ) to compute involve only small values of k (0 and 1). In fact, the approximation is uncanningly accurate: it agrees with the exact value to the first five digits after the decimal point, and is off by only about 0.00008% ! By contrast, normal approximation would be completely

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hw4sol - Math 361 X1 Homework 4 Solutions Spring 2003...

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