Question

Suppose you play rock paper scissors against a computer:

*you gain $1 for each win

*you lose $1 for each lost

if it is a tie, nothing happens

you choose rock 50% of the time and the others 25% of the time

let x be a random variable that represents the amount of money you earn after one game.

(a) find the probability distribution of X

(b) what is your average payoff after 20 games?

(c) what is the standard deviation of the payoff after 20 games?

Answer 1

I am playing rock paper scissors against computer. I choose rock 50% of times and each of papar and scissors 25% of the time i.e. for me the probabilities of choosing rock is 1/2, paper is 1/4 and scissors is 1/4.

Since, nothing has been mentioned we assume computer has the equal probabilities of choosing these three i.e. for computer each probability is 1/3.

Again, if I gain, I win $1. If I lose, I lose $1 and if there is a tie nothing happens.

We are given, X = random variable that represents the amount of money I earn after one game.

Based on this, we prepare the probability distribution table:

I choose	My probability (pm)	Computer chooses	Computer's probability (pc)	Joint Probability = pm x pc	Result	X
Rock	1/2	Rock	1/3	1/6	Tie	0
Rock	1/2	Paper	1/3	1/6	I lose	-1
Rock	1/2	Scissors	1/3	1/6	I win	1
Paper	1/4	Rock	1/3	1/12	I win	1
Paper	1/4	Paper	1/3	1/12	Tie	0
Paper	1/4	Scissors	1/3	1/12	I lose	-1
Scissors	1/4	Rock	1/3	1/12	I lose	-1
Scissors	1/4	Paper	1/3	1/12	I win	1
Scissors	1/4	Scissors	1/3	1/12	Tie	0
Total				1

(a) We combine the probabilities of the similar value of X and get the probability distribution of X is,

(b) The game is being played for 20 times.

My average or expected pay-off after one game is

= 0

Since, each game is independent of the other, therefore the average pay-off after 20 games

= 20 X 0 = 0

Answer: My average pay-off after 20 games is $0.

(c) We calculate the standard deviation of X by using,

We have E(X) = 0

Therefore,

Hence, the standard deviation of the pay-off after 20 games = 20 X 0.8165 = 16.33 (rounded to 2 decimal places)

Answer: The standard deviation of the pay-off after 20 games is $16.33.