Suppose we roll a six-sided die and define the following events. A = get an even

Question

Suppose we roll a six-sided die and define the following events. A = get an even number, so get 2 or 4 or 6 B=get a number less than 3, so get 1 or 2 Are A and B independent? Justify your answer Suppose that 1/3 of new businesses fail in one year. Of those remaining after one year, 1/4 fail in their second year. A. (a) What is the probability that a new business will survive for two years? (b) What is the probability that a new business will fail in its second year? Hint: To fail in the second means you have to have survived the first. C. Suppose events A and B are not independent. For each of the following, say if the statement is true or false. (a) Pr(AIB)Pr(A") (b) Pr(AIP)Pr(A) (c) Pr(AIB")Pr(A*) (d) Pr(A and B) = Pr(A)Pr(B)

Explanation / Answer

Answers below, with explanations:

A. A is event of getting even number
B is event of getting odd number
A and B are independent if P( A and B) = P(A)*P(B)
Since there is no common event between A and B, P( A and B) = 0,
P(A) = P(B) = 1/3
So, P( A and B) != P(A)*P(B)

Hence, A and B aren't independent

B.

a.P( srurvives 2 years) = P( survives both years) = P(survives 1st year)*P(survives 2nd year) = 1/3 * 1/4 = 1/12
b.P( fails 2nd year) = P(survives 1st year)*P(fails 2nd year) = (1/3)(1-1/4) = 1/4

C. We will follow the rule of independence as if A and B are independent then P(A and B) = P(A)*P(B)
So,

a. P(Ac|B) = P(Ac)*P(B)/P(B) = P(Ac)
False
b. P(A|Bc) = P(A)*P(Bc)/P(Bc) = P(A)
False
c. P(Ac||Bc) = P(Ac)*P(Bc)/P(Bc) = P(Ac)
False
d. P(A and B) = P(A)*P(B)
True

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.