A box contains eight balls numbered 1 through 8. Balls 1, 2, and 3 are red; 4, 5

Question

A box contains eight balls numbered 1 through 8. Balls 1, 2, and 3 are red; 4, 5, 6, and 7 are white; 8 is blue. One ball is drawn. Consider the following events

A: The ball is red.

B: The ball is blue.

C: The number is odd.

Which of these events are independent?

A box contains six red and four blue marbles. Two marbles are drawn from the box (without replacement) and their colors are noted. Dene the following events. A: The rst marble is red. B: The second marble is red.

(a) Find P(A); P(A); P(A B); P(B); P(BjA); P(AjB):

(b) Determine of A and B are independent

Explanation / Answer

Question 1:

Here we are given that there are 8 balls numbered from 1 to 8. Therefore 4 have even numbers and 4 got odd numbers. Therefore the probability of various events here are computed as:

P( A) = number of red balls / Total number of balls

P(A) = 3 / 8 = 0.375

Therefore P(A) = 0.375

Also, we have here: 1 of the ball is blue.

Therefore P(B) = 1/8 = 0.125

Also, as half of the balls are odd numbered therefore P(C) = 0.5

Clearly A and B are mutually exclusive events as they cannot happen together and therefore:

P(A and B) = 0

P(A)P(B) = 0.375*0.125 = 0.046875

Therefore A and B are not independent events.

P(A and C) = probability that an odd red ball is drawn = 2/8 = 0.25

Also, P(A)P(C) = 0.375*0.5 = 0.1875 which is not equal to P(A and C)

Therefore A and C are not independent events.

Also, now:

P( B and C) = 2/8 = 0.25

P(B)P(C) = 0.125*0.5 = 0.0625 which is not equal to P(B and C)

Therefore B and C are not independent events.

Therefore none of the events given here are independent.

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