Consider an experiment where 3 balls are drawn from a bin containing 3 red balls

Question

Consider an experiment where 3 balls are drawn from a bin containing 3 red balls and 2 green balls (the balls are not replaced between draws.) Define the events A, B, and C as follows: A = {both balls are red}, B = {both balls are green}, C = {the first ball is red} a) What is the sample space b) What is the probability associated with each of the sample points c) Find the probability of A, B, and C d) Find P(A|B) and P(B|A) e) Arc the events A and B independent, mutually exclusive, or dependent is some other way? g) Are the events A and C independent, mutually exclusive, or dependent in some other way? Consider a fair 6 sided dice where one side is labeled 1, one side is labeled 5, and the others are all labeled 0. Tlhe die is rolled twice. Define the events A, B, and C as follows: A = {The first die is a 1}, B = {The first die is a 5}, C = {The second die is a 0}

Explanation / Answer

4) a) Total sample space = 5C3 = 10

b) P (red ball) = 3 / 5

P (green ball) = 2 / 5

C) P(A) = 3C2 * 2C1 / 5C3 = 3 * 2 / 10 = 3 / 5

P(B) = 2C2 * 3C1 / 5C3 = 1 * 3 / 10 = 3 / 10

P(C) = P(RRG) + P(RGR) + P(RGG) + P (RRR)

= (3 / 5 * 2 /4 * 2 / 3) + (3 / 5 * 2 / 4 * 2 / 3) + (3 / 5 * 2 / 4 * 1 / 3) + ( 3 / 5 * 2 / 4 * 1 / 3)

= 1 / 5 + 1 / 5 + 1 / 10 + 1 / 10 = 6 / 10 = 3 / 5

D) P(A and B) = 0

P(A | B) = P(A and B) / P(B) = 0

P(B | A) = P (B and A) / P (A) = 0

E) P (A and B) = 0

So the events A and B are mutually exclusive.

F) P (A | C) = P( A and C) / P (C) = (P(RRG) + P(RGR)) /( 3 / 5)

= (1/ 5 + 1 / 5) /( 3 / 5 )= (2/5 )/ (3/5 ) = 2 / 3

P(C | A ) = P(C and A) / P(A) = (2/5 )/ (3/5 )= 2 / 3

G) P ( A and C ) = 2 / 5

P (A) * P(C) = 3 / 5 * 3 / 5 = 9 / 25

P(A and C) is not equal to P (A) * P(C)

So they are not independent.

  

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