1. Let S={1,2,3,4}, and A={1,2}, B={1,3}, C={1,4}. Assume the outcomes are equip

Question

1. Let S={1,2,3,4}, and A={1,2}, B={1,3}, C={1,4}. Assume the outcomes are equiprobable. Are A, B and C independent events.
2. Let U be selected at random from the unit interval. Let A={0<U<1/2}, B={1/4<U<3/4}, and C={1/2<U<1}. Are any of these events independent?
3. Let A, B, C be events with probabilities P[A], P[B] and P[C]. (a) Find P[AUB] if A and B are independent (b) Find P[AUB] if A and B are mutually exclusive (c) Find P[AUBUC] if A, B and C are independent (d) Find P[AUBUC] if A, B and C are pairwise mutually exclusive
4. A random experiment is repeated a large number of times and the occurrence of events A and B is noted. How would you test whether events A and B are independent?
5. A fraction p of items from a certain production line is defective (a) What is the probability that there is more than one defective item in a batch of n items?
6. A machine makes errors in a certain operation with probability p. There are two types of errors. The fraction of errors that are type 1 is a, and type 2 is 1-a. (a) What is the probability of k errors in n operations (b) What is the probability of k1 type 1 errors in n operations? (c) What is the probability of k2 type 2 errors in n operations? 1. Let S={1,2,3,4}, and A={1,2}, B={1,3}, C={1,4}. Assume the outcomes are equiprobable. Are A, B and C independent events.
2. Let U be selected at random from the unit interval. Let A={0<U<1/2}, B={1/4<U<3/4}, and C={1/2<U<1}. Are any of these events independent? 2. Let U be selected at random from the unit interval. Let A={0<U<1/2}, B={1/4<U<3/4}, and C={1/2<U<1}. Are any of these events independent?
3. Let A, B, C be events with probabilities P[A], P[B] and P[C]. (a) Find P[AUB] if A and B are independent (b) Find P[AUB] if A and B are mutually exclusive (c) Find P[AUBUC] if A, B and C are independent (d) Find P[AUBUC] if A, B and C are pairwise mutually exclusive 3. Let A, B, C be events with probabilities P[A], P[B] and P[C]. (a) Find P[AUB] if A and B are independent (b) Find P[AUB] if A and B are mutually exclusive (c) Find P[AUBUC] if A, B and C are independent (d) Find P[AUBUC] if A, B and C are pairwise mutually exclusive
4. A random experiment is repeated a large number of times and the occurrence of events A and B is noted. How would you test whether events A and B are independent? 4. A random experiment is repeated a large number of times and the occurrence of events A and B is noted. How would you test whether events A and B are independent?
5. A fraction p of items from a certain production line is defective (a) What is the probability that there is more than one defective item in a batch of n items? 5. A fraction p of items from a certain production line is defective (a) What is the probability that there is more than one defective item in a batch of n items?
6. A machine makes errors in a certain operation with probability p. There are two types of errors. The fraction of errors that are type 1 is a, and type 2 is 1-a. (a) What is the probability of k errors in n operations (b) What is the probability of k1 type 1 errors in n operations? (c) What is the probability of k2 type 2 errors in n operations? 6. A machine makes errors in a certain operation with probability p. There are two types of errors. The fraction of errors that are type 1 is a, and type 2 is 1-a. (a) What is the probability of k errors in n operations (b) What is the probability of k1 type 1 errors in n operations? (c) What is the probability of k2 type 2 errors in n operations?

Explanation / Answer

1.

P(A) = 1/2, P(B) =1/2, P(C) = 1/2

P(AnBnC) = 1/4 ! = P(A)P(B)P(C)

=>

A,B,C are not independent, but they are pairwise independent

2.

P(A) = 1/2, P(B) = 1/2, P(C) = 1/2

P(AnB) = 1/4 = P(A)P(B)

P(AnC) = 0 ! = P(A)P(C)

P(BnC) = 2/4 = P(B)P(C)

=>

A,B are independent

B,C are independent

3.

(a)P(AUB) = P(A)+P(B)-P(A)P(B)

(b)P(AUB) = P(A)+P(B)

(c)P(AUBUC) = P(A)+P(B)+P(C)-P(A)P(B)-P(B)P(C)-P(A)P(C) + P(A)P(B)P(C)

(d)P(AUBUC) = P(A)+P(B)+P(C)

4.

if P(AnB) = P(A)P(B), then they are independent

5.

(a)P = 1-(1-p)^n

6.

(a) nCk (p)^k (1-p)^(n-k)

(b)nCk1 (ap)^k 1(1-ap)^(n-k1)

(c)nCk2 ((1-a)p)^k 2(1-(1-a)p)^(n-k2)

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