Answer to Question #274757 in Statistics and Probability for Maa

Question

You are given the following information on events A, B, C, and D.

P(A) = .4 P(A ∪ D) = .6 P(A ∩ C) = .04

P(B) = .2 P(A⏐B) = .3 P(A ∩ D) = .03

P(C) = .1

a. Compute P(D).

b. Compute P(A ∩ B).

c. Compute P(A⏐C).

d. Compute the probability of the complement of C.

e. Are A and B mutually exclusive? Explain your answer.

f. Are A and B independent? Explain your answer.

g. Are A and C mutually exclusive? Explain your answer.

h. Are A and C independent? Explain your answer.

Answer 1

We are given,

P(A) = 0.4, P(A ∪ D) = 0.6, P(A ∩ C) = 0.04

P(B) = 0.2, P(A⏐B) = 0.3, P(A ∩ D) = 0.03

P(C) = 0.1

a)

Given that, P(A) = 0.4, P(A ∪ D)=0.6, and P(A ∩ D) = 0.03, we can find p(D) using the addition property of probability given as,

P(A ∪ D)=p(A)+p(D)- P(A ∩ D)

From this formula, we can make p(D) subject. Thus, p(D)= P(A ∪ D)+ P(A ∩ D)-p(A).

So,

p(D)=0.6+0.03-0.4=0.23

Therefore, p(D)=0.23

b)

Given that,

P(A⏐B) = 0.3 and P(B) = 0.2, we can determine the value of p(A ∩ B) using the multiplication rule of probability. The multiplication rule of probability is given as,

p(A ∩ B)= P(A⏐B)*p(B)=0.3*0.2=0.06

Therefore, p(A ∩ B)=0.06.

c)

Given that, P(A ∩ C) = 0.04, and p(C)=0.1, we can find the value of p(A|C) using the conditional probability rule given as,

 P(A⏐C)= P(A ∩ C)/p(C)=0.04/0.1=0.4

Therefore,  P(A⏐C)=0.4.

d)

Given that p(C)=0.1, its complement p(C') is given as,

p(C')=1-p(C)=1-0.1=0.9.

Therefore, the  probability of the complement of C, p(C')=0.9

e)

If the events A and B are mutually exclusive then we expect that p(A ∩ B)=0, From part (b) above, we have that $p(A ∩ B)=0.06\not=0$. Since the intersection of events A and B is not equal to 0, then events A and B are not mutually exclusive.

f)

If the events A and B are independent, then we expect that, p(A ∩ B)=p(A)*p(B). from the probabilities given above, p(A ∩ B)=0.06, p(A)=0.4, p(B)=0.2.

Now, p(A)*p(B)=0.4*0.2=0.08.

Clearly,

$p(A ∩ B)\not=p(A)*p(B)$ thus, events A and B are not independent.

g)

If the events A and C are mutually exclusive then we expect that p(A ∩ C)=0. We are given that $p(A ∩ C)=0.04\not=0$. Since the intersection of events A and C is not equal to 0, then events A and C are not mutually exclusive.

h)

If events A and C are independent then we expect that p(A ∩ C) = p(A)*p(C). We are given that, p(A ∩ C) = 0.04, p(A)=0.4 and p(C)=0.1.

Clearly, p(A ∩ C)=0.4*0.1=0.04. Therefore, the condition p(A ∩ C) = p(A)*p(C) is satisfied thus, events A and C are independent.