Question

You play two games against the same opponent. The probability you win the first game is 0.70 If you win the first​ game, the probability you also win the second is 0.50 If you lose the first​ game, the probability that you win the second is 0.20 Complete parts​ a) through​ e).

a. Are the two games independent?

b. What's the probability you lose both games?

c. What's the probability you win both games?

d. Let random variable X be the number of games you win. Find the probability model for X complete the table below (hint: use your answers in part b and c)

e. Find and interpret the expected value of X?

f. What is the standard deviation of X?

Answer 1

a.

Two games are not independent as probability of winning or losing a game is affected by the previous game played.

b.

Probability that you lose both games = Probability that you lose first game * Probability that you lose second game

Probability that you lose first game = 1 - Probability that you win first game = 1- 0.70 = 0.30

Probability that you lose second game = Probability of losing second game given that you lost first game

= 1 - Probability of winning when you lost first game = 1 - 0.20 = 0.80

Probability that you lose both games = Probability that you lose first game * Probability that you lose second game =

= 0.30*0.80 = 0.24

c.

Probability that you win both games = Probability that win lose first game * Probability that you win second game

Probability that you win first game = 0.70

Probability that you win second game = Probability of winning second game given that you won first game = 0.50

Probability that you win both games = Probability that you win first game * Probability that you win second game =

= 0.70*0.50 = 0.35

d.

X can take value 0 , 1 , 2

Probability of winning 0 games = Probability of losing both games = 0.24

Probability of winning 1 game = Probability of winning first game * Probability of losing second game + Probability of losing first game * Probability of winning second game

Here,

Probability of winning first game = 0.70

Probability of losing second game given that you won first game = 1 - Probability of winning second game = 1 -0.50 = 0.50

Probability of losing first game = 1 -0.70 = 0.30

Probability of winning second game given that you lost first game = 0.20

So,

Probability of winning 1 game = 0.70*0.50 + 0.30*0.20 = 0.35+0.06 = 0.41

Probability of winning 2 games = Probability of winning both games = 0.35

So, probability model for X is :

X	0	1	2
P(X=x)	0.24	0.41	0.35

e.

Expected value of X :

x P(X x)=0 0.241 0.41 2 0.35 1.11 E(X) -0,1,2

f.

Standard deviation of X :

2 SD(X) 322P(X - P(X) r-0,1,2 -0,1,2

V02 0.2412 0.4122 0.35 (1.11)2 V1.81 1.112 0.76