collected for use in the approval process required by the
U.S. Food and Drug Administration (FDA). Some participants
were given a placebo, an inert substance that
looks like the drug; others were given the drug. The
data are shown in the following table:
No Help Help
Drug 22 47
Placebo 31 20
aWhat is the probability that the participants perceived that their “medication”
helped if they received the drug?
b. the participants perceived that their “medication”
helped if they received the placebo?
c. the participants perceived that their “medication”
helped?
a. I assume that the "Help" category is "perceived" as helping. Therefore it would be 47 divided by the grand total.
Use the same principle to answer the following two questions.
To calculate the probabilities in this scenario, we need to use the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred.
a. To determine the probability that the participants perceived their "medication" helped if they received the drug, we need to calculate the conditional probability of "Help" given that they received the drug.
P(Help|Drug) = (Number of participants who perceived "Help" and received the drug) / (Total number of participants who received the drug)
In this case, the number of participants who perceived "Help" and received the drug is 47, and the total number of participants who received the drug is 22 + 47 = 69.
P(Help|Drug) = 47 / 69 ≈ 0.68
Therefore, the probability that the participants perceived their "medication" helped if they received the drug is approximately 0.68 or 68%.
b. To determine the probability that the participants perceived their "medication" helped if they received the placebo, we need to calculate the conditional probability of "Help" given that they received the placebo.
P(Help|Placebo) = (Number of participants who perceived "Help" and received the placebo) / (Total number of participants who received the placebo)
In this case, the number of participants who perceived "Help" and received the placebo is 20, and the total number of participants who received the placebo is 31 + 20 = 51.
P(Help|Placebo) = 20 / 51 ≈ 0.39
Therefore, the probability that the participants perceived their "medication" helped if they received the placebo is approximately 0.39 or 39%.
c. To determine the probability that the participants perceived their "medication" helped overall, we can calculate the probability of perceiving "Help" regardless of whether they received the drug or the placebo.
P(Help) = (Number of participants who perceived "Help") / (Total number of participants)
In this case, the number of participants who perceived "Help" is 47 + 20 = 67, and the total number of participants is 22 + 47 + 31 + 20 = 120.
P(Help) = 67 / 120 ≈ 0.56
Therefore, the probability that the participants perceived their "medication" helped overall is approximately 0.56 or 56%.