In the tests of a new pharmaceutical product, data was collected for use in the approval process requied by the U.S. FDA. Some participants were given a placebo, an inert substance that looks like the drug; others were given the drug. The data is shown:

No Help Help
Drug 22 47
Placebo 31 20
What is the probability that
A) the participants perceived that their "medication" helped if they received the drug?
B) the participants perceived that their "medication" helped if they received placebo?
C) the participants perceived that their "medication" helped?

Answer: A) 47/69 68% B) 20/51 39% C) 67/53 14/53 59%

Is this correct?
Thanks.

Notice that your answers to A and B sum to more than 100%. Is that possible?

Your grand total is the sum of all the cells.

A) 47/grand total
B) 20/grand total
C) 67/grand total

(A) and (B) are correct.

I am not sure how your calculated (C).

Question (C) implies favorable results from all participants, namely 22+47+31+20=120.
The favorable results are 47+20=67.
So the percentages are: 67/120 = 56%.

To calculate the probabilities, we need to understand what is being asked and how the data is presented.

In this case, we are interested in the probability of perceiving that the medication helped, given whether the participant received the drug or the placebo.

We can use conditional probability to calculate these probabilities:

A) Probability that participants perceived that their "medication" helped, given they received the drug:
To calculate this probability, we need to use the number of participants who received the drug and reported help, which is 47. We divide this by the total number of participants who received the drug, which is 22 + 47 = 69. Therefore, the probability is 47/69, which is approximately 68%.

B) Probability that participants perceived that their "medication" helped, given they received a placebo:
To calculate this probability, we need to use the number of participants who received the placebo and reported help, which is 20. We divide this by the total number of participants who received the placebo, which is 31 + 20 = 51. Therefore, the probability is 20/51, which is approximately 39%.

C) Probability that participants perceived that their "medication" helped:
To calculate this probability, we add up the total number of participants who reported help, which is 47 (for the drug) + 20 (for the placebo) = 67. We divide this by the total number of participants, which is 22 + 47 + 31 + 20 = 120. Therefore, the probability is 67/120, which simplifies to 14/24 or approximately 59%.

Based on the calculations above, it seems that the provided answers are incorrect. The correct probabilities should be:
A) 47/69 (approximately 68%)
B) 20/51 (approximately 39%)
C) 14/24 or 59%

Note: It's important to double-check the calculations and make sure the data is accurately represented before drawing any final conclusions.