Question

Gambles are independent, and each one results in the playerbeing equally likely to win or lose 1 unit. Let W denote the netwinnings of a gambler whose strategy is to stop gamblingimmediately after his first win. Find

(a) P{W > 0}
(b) P{W < 0}
(c) P E[W]

Answer 1

P{W>0}=0.5

P{W=0}=0.25

E{W}=0

Explanation:

Given:

Gambles are independent i.e. each player is equally likely to win or lose 1 unit. OR each player has equal probability to win or lose 1 unit.

Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win.

Then

A.P{W>0}=?

P{W>0}=0.5, because each player is equally likely to win or lose on first gamble. i.e there is equal chances for winning or losing on the first gamble.

B.P{W<0}=?

for P{W<0} we need to find P{W=0} first as;

P{W=0}=0.25

As there is equal probability to win or lose, after first win, if you want to finish gamble with no profit (equal number of lose and win) then if you losing, you have equal probability to win or lose so to finish your game with P=0 your probability is 0.25 (half of 0.5)

P{W<0} means net lose which is equal to total probability minus probability of profit and probability of net profit equal to zero.

i.e. P{W<0}=1-P{W=0}-P{W>0}

P{W<0}=1-0.25-0.5=0.25

C.E{W}=?

E{W}=P{W>0}*{W>0}+P{W<0}*{W<0}+P{W=0}*{W=0}

E{W}=0.5*(1)+0.5(-1)+0.25*(0) (for any value of W,P{W>0}*{W>0}+P{W<0}*{W<0}=0, because sum of same positive and negative numbers is zero)

E{W}=0