Frequently Asked Questions In Quantitative Finance

Chapter 10: Brainteasers 365

In the balls example,Ais the event that all of the balls
inthe bag are white,Bis the event that the balls
taken out ofthe bag are all white. We want to find
Prob(A|B).

Clearly, Prob(A)isjust^1210 = 0 .000976563. Trivially
Prob(B|A) is 1. The probability that we take 10 white
balls out of the bag is a bit harder. We have to look at
the probability of havingnwhite balls in the first place
and then picking, after replacement, 10 white. This is
then Prob(B). It is calculated as

∑^10

n= 0

10!

n!(10−n)!

1

210

(n

10

) 10

= 0. 01391303.

And so the required probability is 0. 000976563 / 0. 01391303
= 0 .0701905. Just over 7%.

Sums of uniform random variables

The random variablesx 1 ,x 2 ,x 3 ,...are independent and
uniformly distributed over zero to one. We add upnof
them until the sum exceeds one. What is the expected
value ofn?

(Thanks to balaji.)

Solution
There are two steps to finding the solution. First what
is the probability of the sum ofnsuch random vari-
ables being less than one. Second, what is the required
expectation.

There are several ways to approach the first part. One
way is perhaps the most straightforward, simply calcu-
late the probability by integrating unit integrand over