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To find P(placebo and improvement), we need to first find the probability of receiving a placebo and then the probability of reporting an improvement if receiving a placebo.
P(placebo) = 1 - P(medication A) = 1 - 0.80 = 0.20
P(no improvement | placebo) = 0.62
P(improvement | placebo) = 1 - P(no improvement | placebo) = 1 - 0.62 = 0.38
Now, we can find P(placebo and improvement) using the formula for conditional probability:
P(placebo and improvement) = P(improvement | placebo) * P(placebo)
P(placebo and improvement) = 0.38 * 0.20
P(placebo and improvement) = 0.076
Therefore, the probability of receiving a placebo and reporting an improvement is 0.076 or 7.6%.