370 Frequently Asked Questions In Quantitative Finance
The answer to the first question is then what is the
probability of getting 165 or more ‘wins’ out of 260
when the probability of a ‘win’ is 0.6. The answer to this
standard probability question is just over 14%.
The average return per day is
1 −exp(0.6ln1. 5 + 0 .4ln0.5)=− 3 .34%.
The probability of the trader making money after one
day is 60%. After two days the trader has to win on
both days to be ahead, and therefore the probability is
36%. After three days the trader has to win at least two
out of three, this has a probability of 64.8%. After four
days, he has to win at least three out of four, probability
47.52%. And so on. With an horizon ofNdays he would
have to win at leastNln 2/ln 3 (or rather the integer
greater than this) times. The answer to the second part
of the question is therefore three days.
As well as being counterintuitive, this question does
give a nice insight into money management and is
clearly related to theKelly criterion. If you see a ques-
tion like this it is meant to trick you if the expected
profit, here 0. 6 × 0. 5 + 0. 4 ×(− 0 .5)= 0 .1, is positive
with the expected return, here− 3 .34%, negative.
Dice game
You start with no money and play a game in which you
throw a dice over and over again. For each throw, if 1
appears you win $1, if 2 appears you win $2, etc. but if
6 appears you lose all your money and the game ends.
When is the optimal stopping time and what are your
expected winnings?
(Thanks to ckc226.)
