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

Answer:

Given Data,

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).

The Natasha agree to play flip a coin 5 time. Possible outcomes are,

(i) 5 heads ,0 tails

(ii) 4 heads, 1 tails

(iii) 3 heads,2 tails

(iv) 2 heads, 3 tails

(v) 1 head, 4 tails

(vi) 0 heads,5 tails

S=Number of H-Number of tails.

Hence,probability distinct of S

S	-5	-4	-3	-2	-1	0	1	2	3	4	5
P(S=s)	1/32	0	5/32	0	10/32	0	10/32	0	5/32	0	1/32

Here, X Bin(5,1/2).

P(X=n)=(5_x)(1/2)⁵

Here X=Number of Heads

P(S=-5)=P(0 head,5 tails):

P(X=0)=1/32

P(x=5)=P(S=5)

P(S=-3)=P(1 head,4 tails):

P(X=1)=(5₁)(1/2)⁵=5/32

P(X=4)=P(S=3)

P(S=-3)=P(2 head,3 tails):

P(X=2)=10/32

P(X=3)=P(S=1)

Therefore, remember that the binomial distribution is symmetric when p=1/2.

Distribution of S

S	-5	-3	-1	1	3	5
P(S=s)	1/32	5/32	10/32	10/32	5/32	1/32

The value multiplied by corresponding probability.

Graph of Natasha's Payoff:

(b).

On an average Natasha is willing to give Boris no amount of money as the expectation represents.

Best case for Natasha is 5 Heads,0 Tails.

Worst case for Natasha is 0 Heads, 5 Tails

(c).

Here,Game slightly changes Natasha is gainer only, she have to pay nothing now.

S=0,1,3,5

Now, P(S=0)

=P[(2H,3T),(1H,4T),(0H,5T)]

=16/32=1/2

P(S=1)=10/32

P(S=3)=5/32

P(X=5)=1/32

Hence now,

Graph:

This graph remains that now Natasha is only gainer she have total probability in her favour that she will lose with 0 probability.

(d).

Natasha will gain 15/16 amount of money on an average . In the last case Natasha will have to pay nothing since she have no chance of losing.

So, now best outcome for her is 5 head and no tails and worst outcome is all the outcomes where number of heads is less than number of tails since in those outcomes she will gain nothing