364 Frequently Asked Questions In Quantitative Finance
This is because with probability^12 we get the required
head, and with probability^12 we get a tail and will have
to start the whole thing anew. Therefore we obtain
Nn= 2 Nn− 1 + 2.
This has solution
Nn= 2 n+^1 − 2.
This means six tosses on average to get two heads in
arow.
Balls in a bag
Ten balls are put in a bag based on the result of the
tosses of an unbiased coin. If the coin turns up heads,
put in a black ball, if tails, put in a white ball. When the
bag contains ten balls hand it to someone who hasn’t
seen the colours selected. Ask them to take out ten
balls, one at a time with replacement. If all ten examined
balls turn out to be white, what is the probability that
all ten balls in the bag are white?
(Thanks to mikebell.)
Solution
This is a test of your understanding of conditional prob-
ability and Bayes’ theorem again. First a statement of
Bayes’ theorem.
Prob(A|B)=
Prob(B|A)Prob(A)
Prob(B)
.
Prob(A) is the probability ofA, without any information
aboutB, this is unconditional probability. Prob(A|B)
means probability ofAwith the extra information
concerningB, this is a conditional probability.
