Question

6. If ?(?) = 0.5 and ?(?|?) = 0.4, are events ? and ? independent? Why or why not?

7. A report on addiction notes that 16% of Americans have the disease of addiction, and 90% of them receive no form of treatment. Determine the probability that a randomly selected American has the disease of addiction and receives no form of treatment.

Answer 1

Solution:

Question 6)

Given:?(?) = 0.5 and ?(?|?) = 0.4

We have to find if events ? and ? independent or not.

Events ? and ? independent if and only if:

P( B | A ) = P( B )

Since P( B | A ) = 0.4 and P(B) = 0.5

thus P( B | A ) P( B )

Thus events ? and ? are not independent.

Question 7)

Given:

P( American has the disease of addiction) = 16% = 0.16

and

90% of them receive no form of treatment

Thus we have:

P( Receive no form of treatment given that American has the disease of addiction) = 0.90

Let A = American has the disease of addiction and B = Receive no form of treatment

thus we have:

P(A) = 0.16 and P( B | A) = 0.90

We have to determine the probability that a randomly selected American has the disease of addiction and receives no form of treatment.

That is:

P( American has the disease of addiction and receives no form of treatment)= .............?

That is:
P( A and B) = ............?

Using conditional probability formula we get:

that is:

Thus the probability that a randomly selected American has the disease of addiction and receives no form of treatment is 0.144