Problem 1 [22 pts (4,4,6,4,4)]: Black and White An urn contains 10 balls: 6 whit

Question

Problem 1 [22 pts (4,4,6,4,4)]: Black and White An urn contains 10 balls: 6 white balls numbered 1 through 6, and 4 black balls numbered 1 through 4. We are simultaneously and randomly drawing 2 balls out of the 10. 1. Find the probability of event A: the two balls are white 2. Find the probability of event B:the two balls are odd 3. Are events A and B independent? Prove your answer 4. Let X be the random variable whose value is the number of white balls in the drawing (a) Write the probability distribution of X in form of a table: PrX0 Pr[X-1] Pr[X: b) Find the expected value EX

Explanation / Answer

a) Probability that both the balls are white is computed as:

= (6/10)*(5/9) = 0.3333

Therefore 0.3333 is the required probability here.

b) Probability that both the balls are odd numbered is computed as:

= (5/10)*(4/9)

= 0.2222

Therefore 0.2222 is the required probability here.

c) Probability that both the balls are odd and white is computed as:

P(W and odd) = (3/10)*(2/9) = 0.0667

P(W)P( odd) = 0.3333*0.2222 = 0.0741

Therefore as P(W and odd) is not equal to P(W)P(odd), therefore the 2 events are not independent

d) The PDF for X here is given as:

The expected value of X now is computed as:

E(X) = 0.1333*0 + 0.5333*1+ 0.3333*2 = 1.20

Therefore 1.20 is the expected value of X here.

P(X = 0) P(X = 1) P(X = 2) (4/10)*(3/9) = 0.1333 (4/10)*(6/9) + (6/10)*(4/9) = 0.5333 (6/10)*(5/9) = 0.3333

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