Question

4 multiple choice 10 points
which condition is necessary for the multiplication rule to apply?

• events a and b must be independent
• event a must occur first
• event b must occur first
• events a and b must be mutually exclusive

5 multiple choice 10 points
in a clinical trial, 30% of patients experience nausea as a side effect of a medication. among those who experience nausea, 40% also develop headaches. which expression correctly represents the probability that a randomly selected patient will experience both nausea and headaches?

• $0.30 \times 0.40$
• $0.30 + 0.40$
• $0.40 \times p(\text{nausea} | \text{headache})$
• $0.30 + 0.40 - (0.30 \times 0.40)$

6 multiple choice 10 points
a diagnostic test for a disease has a false positive rate of 5% and a false negative rate of 2%. the disease prevalence is 10%. if a person tests positive, what is the probability they actually have the disease?

• $.5$
• $.10$
• $.68$
• $.95$

Explanation:

Question 4

The multiplication rule for probability: For independent events, \( P(A \cap B)=P(A)\times P(B) \). For dependent events, \( P(A \cap B)=P(A)\times P(B|A) \) (or vice versa). The key for the basic multiplication rule (when using \( P(A)\times P(B) \)) is independence. Mutually exclusive events have \( P(A \cap B) = 0 \), so multiplication rule (as \( P(A)\times P(B) \)) doesn't apply. Order of occurrence (A first or B first) isn't a condition for the multiplication rule. So the correct condition is that events A and B must be independent.

Let \( N \) be the event of nausea and \( H \) be the event of headache. We know \( P(N)=0.30 \) and \( P(H|N)=0.40 \). The formula for the probability of both events (intersection) is \( P(N \cap H)=P(N)\times P(H|N) \). Substituting the values, we get \( P(N \cap H)=0.30\times0.40 \). The addition rule is for union, not intersection, so \( 0.30 + 0.40 \) is wrong. The other options don't match the conditional probability formula for intersection.

Step 1: Define Events

Let \( D \) be the event that a person has the disease, and \( + \) be the event that a person tests positive. We know:

• Prevalence (probability of having the disease): \( P(D) = 0.10 \)
• Probability of not having the disease: \( P(

eg D)=1 - P(D)=1 - 0.10 = 0.90 \)

• False positive rate: \( P(+|

eg D)=0.05 \) (tests positive when no disease)

• False negative rate: \( P(

eg +|D)=0.02 \), so \( P(+|D)=1 - P(
eg +|D)=1 - 0.02 = 0.98 \)

Step 2: Apply Bayes' Theorem

Bayes' Theorem states \( P(D|+)=\frac{P(+|D)\times P(D)}{P(+|D)\times P(D)+P(+|
eg D)\times P(
eg D)} \)

Substitute the values:

• \( P(+|D)\times P(D)=0.98\times0.10 = 0.098 \)
• \( P(+|

eg D)\times P(
eg D)=0.05\times0.90 = 0.045 \)

• Denominator: \( 0.098 + 0.045 = 0.143 \)
• Then \( P(D|+)=\frac{0.098}{0.143}\approx0.68 \)

Answer:

A. Events A and B must be independent

Question 5