A company is studying the effects of a new medication. In this study, 70% of the participants received the new medication while the other participants received a placebo. Of those who received the new medication, 80% reported an improvement. Of those who received the placebo, 90% reported no improvement. Based on the results of the study, how much greater is the probability of a person reporting improvement and taking the new medication than the probability of a person reporting improvement and taking the placebo?
To calculate the probability of a person reporting improvement and taking the new medication, we first need to calculate the overall probability of improvement:
P(improvement) = P(improvement | new medication) * P(new medication) + P(improvement | placebo) * P(placebo)
P(improvement) = 0.80 * 0.70 + 0.10 * 0.30
P(improvement) = 0.56 + 0.03
P(improvement) = 0.59
Now, we can calculate the probability of a person reporting improvement and taking the placebo:
P(improvement | placebo) = 0.10
To find out how much greater the probability of improvement is for someone taking the new medication compared to someone taking the placebo, we can calculate the difference:
P(improvement | new medication) - P(improvement | placebo) = 0.80 - 0.10 = 0.70
Therefore, the probability of a person reporting improvement and taking the new medication is 0.70 greater than the probability of a person reporting improvement and taking the placebo.