Question

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

Rearranging the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) to solve for \( P(B) \), we get \( P(B) = P(A \cup B) - P(A) + P(A \cap B) \).

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

Step4: Calculate the value of \( P(B) \)

First, \( 0.08 - 0.05 = 0.03 \), then \( 0.03 + 0.02 = 0.05 \)? Wait, no, wait: Wait, \( 0.08 - 0.05 + 0.02 = 0.05 \)? Wait, no, let's do it again. 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 \( 0.08 - 0.05 + 0.02 = 0.05 \)? Wait, no, 0.08 - 0.05 is 0.03, plus 0.02 is 0.05? Wait, that can't be. Wait, no, maybe I made a mistake. 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, but let's check again. Wait, if \( P(A)=0.05 \), \( P(A\cap B)=0.02 \), \( P(A \cup B)=0.08 \). Then \( P(A \cup B)=P(A)+P(B)-P(A\cap B) \) => \( 0.08 = 0.05 + P(B) - 0.02 \) => \( 0.08 = P(B) + 0.03 \) => \( P(B)=0.08 - 0.03 = 0.05 \)? Wait, no, 0.05 + P(B) - 0.02 = 0.08 => P(B) + 0.03 = 0.08 => P(B) = 0.05? Wait, but that would be 5 percent. But let's check the options. Wait, the options are 1,5,11,15. Wait, maybe I messed up. Wait, no, let's do it again. Let's use the formula correctly. The formula for the union of two events is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). So we need to find \( P(B) \). So rearranged, \( P(B) = P(A \cup B) - P(A) + P(A \cap B) \). So plugging in the numbers: \( P(A \cup B) = 0.08 \), \( P(A) = 0.05 \), \( P(A \cap B) = 0.02 \). So \( P(B) = 0.08 - 0.05 + 0.02 = 0.05 \), which is 5 percent. Wait, but that's one of the options (5 percent). Wait, but let's verify. 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. So the probability of experiencing heartburn is 5 percent.

Answer:

5 percent (the option is "5 percent")