Question

1.Mental health. The National Survey on Drug Use and Health reports that 18.1% of all adults in the United States had a mental illness in 2014. Among adults with a substance use disorder, 39.1% had a mental illness. By comparison, only 16.2% of adults without a substance use disorder had a mental illness. The report also states that 3.3% of American adults had both a mental illness and a substance use disorder. Use the notation MI and SUD for mental illness and substance abuse disorder, respectively.

1. Express the four percents cited here as probabilities for a randomly selected American adult. Use proper probability notation.

2. Obtain the probability P(SUD|MI). Write a sentence reporting this probability in context.

3. Explain in context why P(SUD|MI) and P(MI|SUD) are not the same thing.

Answer 1

Solution-

(a)

According to question,

Notation MI and SUD used for for mental illness and substance abuse disorder, respectively.

Since 18.1% of all adults in the United States had a mental illness.

So, P(MI) = 18.1/100 = 0.181

Since among adults with a substance use disorder, 39.1% had a mental illness.

It means percentage of adults having mental illness if it is given that they have substance use disorder is 39.1%.

So, P(MI/SUD) =39.1/100 = 0.391

Since 16.2% of adults without a substance use disorder had a mental illness.

It means percentage of adults having mental illness if it is given that they are without a substance use disorder is 16.2%.

So, P(MI/SUD')= 16.1/100 = 0.161

(Here, SUD' denotes adults without a substance disorder).

Since 3.3% of American adults had both a mental illness and a substance use disorder. So,

Hence, the four percents as probabilities for a randomly selected American adult are

P(MI) = 0.181

P(MI/SUD) = 0.391

P(MI/SUD') = 0.162

(b) The probability P(SUD|MI) is given by Conditional Probability as

On putting the values(from part a), we get

Hence, P(SUD/MI) =0.1823

It means the probability that the selected adult is with substance use disorder if it is given that he/she has mental illness is 0.1823

​​​​​​(c)

As calculated above-

P(SUD/MI) = 0.1823

P(MI/SUD) = 0.391

P(SUD|MI) and P(MI|SUD) are not the same thing because

P(SUD/MI) represents the probability that the selected individual is with substance use disorder if it is given that he/she has mental illness.

P(MI/SUD) represents the probability that the selected adult have mental illness if it is given that adult is with substance use disorder.

It means P(SUD/MI) is probability of SUD with given condition of MI and P(MI/SUD) is probability of MI with given condition of SUD.

So, both are conditional probability with different probability and conditions.

Hence, P(MI/SUD) and P(SUD/MI) are not the same thing.