Question

calculating a probability
a pharmaceutical company collects data on the side effects participants report for a drug it is testing. the company finds that the probability of experiencing a headache is 5 percent and the probability of experiencing both a headache and heartburn is 2 percent. if the probability of experiencing a headache or heartburn is 8 percent, what is the probability of experiencing heartburn?
○ 1 percent
○ 5 percent
○ 11 percent
○ 15 percent

Explanation:

Step1: Recall the formula for the probability of the union of two events

The formula for the probability of the union of two events \( A \) and \( B \) is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( P(A \cup B) \) is the probability of \( A \) or \( B \), \( P(A) \) is the probability of \( A \), \( P(B) \) is the probability of \( B \), and \( P(A \cap B) \) is the probability of \( A \) and \( B \).

Let \( A \) be the event of experiencing a headache and \( B \) be the event of experiencing heartburn. We know \( P(A) = 0.05 \), \( P(A \cap B) = 0.02 \), and \( P(A \cup B) = 0.08 \). We need to find \( P(B) \).

Step2: Rearrange the formula to solve for \( P(B) \)

From \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), we can rearrange it to solve for \( P(B) \):

\( P(B) = P(A \cup B) - P(A) + P(A \cap B) \)

Step3: Substitute the known values into the formula

Substitute \( P(A \cup B) = 0.08 \), \( P(A) = 0.05 \), and \( P(A \cap B) = 0.02 \) into the formula:

\( P(B) = 0.08 - 0.05 + 0.02 \)

\( P(B) = 0.03 + 0.02 \)

\( P(B) = 0.05 \)? Wait, no, wait: \( 0.08 - 0.05 = 0.03 \), then \( 0.03 + 0.02 = 0.05 \)? Wait, that can't be right. Wait, no, let's check again. Wait, the formula is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), so solving for \( P(B) \):

\( P(B) = P(A \cup B) - P(A) + P(A \cap B) \)

So \( 0.08 - 0.05 + 0.02 = 0.05 \)? Wait, 0.08 - 0.05 is 0.03, plus 0.02 is 0.05? Wait, but that would be 5 percent. But let's check again. Wait, maybe I made a mistake. Wait, the formula is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), so \( P(B) = P(A \cup B) - P(A) + P(A \cap B) \). So 0.08 (which is 8 percent) minus 0.05 (5 percent) is 0.03 (3 percent), then plus 0.02 (2 percent) is 0.05 (5 percent). So \( P(B) = 0.05 \), which is 5 percent. Wait, but let's verify with the original formula. If \( P(A) = 0.05 \), \( P(B) = 0.05 \), \( P(A \cap B) = 0.02 \), then \( P(A \cup B) = 0.05 + 0.05 - 0.02 = 0.08 \), which matches the given \( P(A \cup B) = 0.08 \). So that works.

Wait, but the options are 1 percent, 5 percent, 11 percent, 15 percent. So 5 percent is an option. So the probability of experiencing heartburn is 5 percent? Wait, but let me check again. Wait, maybe I messed up the formula. Wait, the formula for the union of two events is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). So solving for \( P(B) \):

\( P(B) = P(A \cup B) - P(A) + P(A \cap B) \)

So 0.08 (8%) - 0.05 (5%) + 0.02 (2%) = 0.05 (5%). So that's 5 percent. So the answer is 5 percent.

Wait, but let me check again. If \( P(A) = 0.05 \), \( P(B) = 0.05 \), \( P(A \cap B) = 0.02 \), then \( P(A \cup B) = 0.05 + 0.05 - 0.02 = 0.08 \), which is correct. So that works. So the probability of heartburn is 5 percent.

Answer:

5 percent (Option B: 5 percent)