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,
nonprogrammable 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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 Spring '03
 aldous
 Normal Distribution, Probability theory, Binomial distribution, 5 %, 2 2k

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