Question

1. Consider a bet with 2.1.1 odds. This means that if you wager $1, you get back $2.10 if you win and $0 if you lose. Assume you start with $1 and place a single $1 bet. Let the RV X denote the change in your final wealth, i.e., P(X € (-1,1.1}) = 1. (a) (2 points) If the probability of winning is 45%, find your expected change in wealth E(X). (b) (2 points) If the probability of winning is 45%, would you take the bet? Justify your answer. (c) (3 points) If the probability of winning is 55%, what's E(X)? Would you take the bet? Justify your answer. (d) (3 points) What is the minimum value of the probability of winning that would make you take the bet? Justify your answer. (e) (0 points) Assume the probability of winning is 55%, but you have to wager your entire life's savings. Would you take this bet? What if you were Goldman Sachs placing a $1,000,000 bet? Justify your answer. 10 are in a rush, so you place

Answer 1

Sol 1 Rv, denetes the chense in finel welth -1 (x-x)055 0.45 E(x) -xo.55 1-1 * 6 45 o 055 No beccme my exprcted ehenge in L ney tive (-o.055) X- r(x-*) 45 F(x) = -1x 0-45 + 1.1 xo-55 O.156 Since, my exy ected chenje in inel eelth Psitie I would tke thr bet Let tie minimum velne it mbibilt winnig-h -1 f(x-X) 1-P E (x) -x (1-h) + 1-1X I winld neilt 2.1X teke te be it my rxITet td chenge n puikve or ltrst eaeergies 70) E(X) 70 2-1xP -17 ) 0 476 inimum v I bt Prbebihty t winning o.476