(Mathematical statistics) Hi, would you mind answering the question? Thank you s

Question

(Mathematical statistics)

Hi, would you mind answering the question?

Thank you so much.

Suppose you have two urns. At the beginning of the experiment, Urn 1 contains three yellow balls, three red balls, and three green balls. Also, Urn 2 contains one yellow ball, two green balls and four purple balls. Consider a two-stage experiment in which we randomly draw three balls from Urn 1 and move them to Urn 2, and then we randomly draw one ball from the updated Urn 2. a. Define two events as follows: A = {Two yellow balls and one green ball are moved to Urn 2} and B = A green ball is drawn from Urn 2 Find the probabilities of these two events Are A and B independent? Find the probability that at least two of the balls moved from Urn 1 to Urn 2 were yellow, given that the ball drawn from Urn 2 was yellow. b. c.

Explanation / Answer

a. Here A is definded as such

Pr(A) = Pr(we select 2 yellow balls and one green ball out of three yellow, three red and three green balls) =

3C2 * 3C0 * 3C1/ 9C3 = 3 * 3/ 84 = 9/84 = 3/28

Pr(B) = Pr(A green ball is drawn from urn 2) = Pr(0 Green ball drawn from urn 1) * Pr(1 green ball out of the 10 (7 + 3) balls from urn2) + Pr(1 green ball from urn 1) * Pr(1 green ball out of the 10 (7 + 3) balls from urn2) + Pr(2 green ball from urn 1) * Pr(1 green ball out of the 10 (7 + 3) balls from urn2) + Pr(3 green ball from urn 1) * Pr(1 green ball out of the 10 (7 + 3) balls from urn2)

= 6C3 * 3C0 /9C3 * (2/10) + 6C2 * 3C1 /9C3 * (3/10) + 6C1 * 3C2 /9C3 * (4/10) + 6C0 * 3C3 /9C3 * (5/10)

= (20/84) * 0.2 + (45/84) * 0.3 + (18/84) * 0.4 + (1/84) * 0.5

= 0.3

(b) Here if A and B are independent, then

Pr(A and B) = Pr(A) * Pr(B)

So, Pr(A) = 3/28 and Pr(B) = 0.3

Pr(A and B) = Pr(two yellow balls and one green balls are moved to urn 2 and A green ball is drawn from urn 2)

= 3C2 * 3C0 * 3C1 /9C3 * (3/10) = (20/84) * 0.3 = 0.0714

Pr(A and B) = Pr(A) * Pr(B)

so both events are independent in nature.

(c) Here we have to find that at least two of the balls moved from urn 1 to urn 2 were yellow, given that ball drawn from urn 2 was yellow.

Here,

Pr(Ball drawn from urn 2 was yellow) = Pr(0 yellow ball drawn from urn 1) * Pr(1 yellow ball out of the 10 (7 + 3) balls from urn2) + Pr(1 yellow ball from urn 1) * Pr(1 yellow ball out of the 10 (7 + 3) balls from urn2) + Pr(2 yellow ball from urn 1) * Pr(1 yellow ball out of the 10 (7 + 3) balls from urn2) + Pr(3 yellow ball from urn 1) * Pr(1 yellow ball out of the 10 (7 + 3) balls from urn2)

=  6C3 * 3C0 /9C3 * (1/10) + 6C2 * 3C1 /9C3 * (2/10) + 6C1 * 3C2 /9C3 * (3/10) + 6C0 * 3C3 /9C3 * (4/10)

= 0.2

so,

here Pr(at least two or more yellow balls are drawn from urn 1 l Ball drawn from urn 2 was yellow) =

[6C1 * 3C2 /9C3 * (3/10) + 6C0 * 3C3 /9C3 * (4/10) ]/ 0.2 = (18/84 * 0.3 + 1/84 * 0.4)/0.2 = 0.3452

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