Question

Boris and Natasha agree to play the following game. They will flip a (fair) coin 5 times in a row. They will compute S = (number of heads H – number of tails T).

a) Boris will pay Natasha S. Graph Natasha’s payoff as a function of S. What is the expected value of S?

b) How much should Natasha be willing to pay Boris to play this game? After paying this amount, what is her best case and worst case outcome?

This time, after 5 flips of the coin, if there are more heads H than tails T, Boris will pay Natasha H – T. If there are more tails T than heads H, Boris will pay Natasha nothing.

c) Graph Natasha’s payoff as a function of S = H – T. What does this graph remind you of?

d) What is the expected value of Natasha’s payoff? How much should she be willing to pay to play this game? After paying this amount, what is her best case and worst case outcome?

Answer 1

AnsweL: The Natasha agace to play flip a coin 5 times. @ pay Natasha s qolaph Natasha payoft of 5 tosses a coin possible outcomes. (i) 5 heads o tails (i) 4 heads, 1 taids (1°) 3 heads, 2 taids tiu) 2 heads i 3 fails (NI I head , 4 taids (vi) o head, 5 taids, s = Number of H - Number of tails probability distoribution of s S -5 -4 -3 -2 -1 0 1 2 3 4 5 P(Sza) o çoi o oso Š x no Bin (5,1/2) P(x=0 ) - ( 5x ) ()5 No. of heads x P (5=-5)- P (o head, 5 tails) 32

= P(x=0): 32 = P(+5) =P (5:5) P (s=-3)= pllhead, it tails/ .P (=1 ) - ( ) (2)5 S EP (n=4) - P(5-3) P (5=-1) = P (2 head, 3 taids) = P (x=2)= 10 P(x = 3) - P(5=1) The Binomial distoibution sy metod paz Disteribution of S S - 5 -3 - 1 3 5 - P(S=s) 1 32 15 12 Lo si ho o 5 1 2 2 2 2 E-5 32 -15 32 B -10 + 10 + 15 +15 = 0 32 32 32 multiplied by corresponding The value poro bability

45 -3 -1 0 1 3 5 The paying this amount times flips average ratasha willing Boris amount money Expectations respects best Case Matosha 5 head o tail worst case Natasha o head 5 tails © Graph Natasha payoff. SEHAT game slight change S=0 1,3,5 P(0:0) = Pſ(2+, 3t ), (14,47) (04,5T)] - 32 p (s = ) - 10 P(5-3)2. P (S=5). z e(s) = (* *0)+ (**1)+(* 3)+(* *5)

- eli tol to 0 1 3 5 This graph cremainds Natasha only gainless total porobability will lose with probability. Natasha gain 15 amount on an average lost case pay nothing no chance of losing out come hear s head and tail s no out come all the outcome no of heads less those outcome will again nothing