You play two games against the same opponent. The probability you win the first

Question

You play two games against the same opponent. The probability you win the first game is 0.4. If you win the first game, the probability you also win the second one is 0.2. If you lose the first game, the probability that you win the second game is 0.3.

a. Are the two games independent?

Explain your answer.

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)

X

P(x)

0

1

2

e. Find and interpret the expected value of X?

f. What is the standard deviation of X?

X

P(x)

0

1

2

Explanation / Answer

a) No, Since the probability of outcomes of second game is dependent on the first game.

b) P(lose first and lose second) = 0.6*0.7 = 0.42

c) P(win first and win second) = 0.4*0.2 = 0.08

d)

P(1) = 1 - (P(0) + P(2)) = 1-(0.42+0.08) = 0.5

e) E(X) = X*P(X) = 0*0.42 + 0.5*1 + 0.08*2 = 0.66, Expected to win 0.66 games, it doesnt make much sense for one game, but for large number of these games, the average matches one will tend 0.66

f) Var(X) = (X-E(X))2*P(X) = (0-0.66)2*0.42 + (1-0.66)2*0.5 + (2-0.66)2*0.08 = 0.3844,
std. dev. = sqrt(0.3844) = 0.62

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