Question

In: Statistics and Probability

14. There are eight balls in an urn, three red labeled {1, 2, 3}, a blue...

14. There are eight balls in an urn, three red labeled {1, 2, 3}, a blue ball marked {4}, and four white balls marked {5, 6, 7, 8}. Given the following three events, tell whether each possible pair of events (A and B, A and C, and B and C) is independent, disjoint, or neither. A. A red ball is drawn. B. An even numbered ball is drawn. C. A white ball is drawn.

Solutions

Expert Solution

Two events are disjoint of they both cannot occur at the same time.

Two events A and B are independent if P(A) x P(B) = P(A and B)

1. Consider A and B,

It is possible for a ball to be both red and even numbered (Red ball labelled 2). Therefore, the events are not disjoint

P(A) = 3/8

P(B) = 4/8 = 1/2

P(A) x P(B) = 3/8 x 1/2 = 3/16

P(A and B) = 1/8

P(A) x P(B) P(A and B)

Therefore A and B are not independent

Ans: A and B are not disjoint and not independent

2. Consider A and C,

A ball cannot be both red and white at the same time.'

Therefore, A and C are disjoint

P(A and C) = 0

P(A) = 3/8

P(C) = 1/2

Ans: A and C are disjoint and not independent

3. Consider B and C,

It is possible for a ball to be both white and even numbered (White balls labelled 6 and 8). Therefore, the events are not disjoint

P(B) = 4/8 = 1/2

P(C) = 4/8 = 1/2

P(B) x P(C) = 1/2 x 1/2 = 1/4

P(B and C) = 2/8 = 1/4

P(B) x P(C) = P(B and C)

Therefore A and B are independent

Ans: B and C are not disjoint and are independent.


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