the resultant of A and B makes an angel alpha with A and bets with B, then
Answers
Answer:
Explanation:
Random walk and Gambler’s ruin. Imagine a walker moving along a line. At every
unit of time, he makes a step left or right of exactly one of unit. So we can think that his
position is given by an integer n ∈ Z. We assume the following probabilistic rule for the
walker starting at n
if in position n move to n + 1 with probability p
move to n − 1 with probability q
stay at n with probability r (1)
with
p + q + r = 1
Instead a walker on a line you can think of a gambler at a casino making bets of $1
at certain game (say betting on red on roulette). He start with a fortune of $n. With
probability p he doubles his bet, and the casino pays him $1 so that he increase his fortune
by $1. With probability q he looses and his fortune decreases by $1, and with probability
r he gets his bet back and his fortune is unchanged.
As we have seen in previous lectures, in many such games the odds of winning are very
close to 1 with p typically around .49. Using our second order difference equations we will
show that even though the odds are only very slightly in favor of the casino, this enough
to ensure that in the long run, the casino will makes lots of money and the gambler not so
much. We also investigate what is the better strategy for a gambler, play small amounts
of money (be cautious) or play big amounts of money (be bold). We shall see that being
bold is the better strategy if odds are not in your favor (i.e. in casino), while if the odds
are in your favor the better strategy is to play small amounts of money.
We say that the game is
fair if p = q
subfair if p < q
superfair if p > q
The gambler’s ruin equation: In order to make the previous problem precise we
imagine the following situation.
• You starting fortune if $j.
• In every game you bet $1