SOLUTION: Many e-mail account holders get spam e-mail inviting the purchase of certain “erectile assistance” drugs like Cialis and Tadalfil. (I get these regularly). Such drugs have to b

Question 1166559: Many e-mail account holders get spam e-mail inviting the purchase of certain “erectile assistance” drugs like Cialis and Tadalfil. (I get these regularly). Such drugs have to be approved by the FDA in USA before sale. Suppose there is a 0.95 chance the FDA will approve a new drug if the results of current testing show no side effects; but they will also approve the drug with a 0.50 chance if testing shows side effects. A certain physician working for the drug manufacturer believes there is a 0.20 chance the drug will cause side effects. What is the probability the drug will be approved by the FDA?
Answer by CPhill(2030) (Show Source): You can put this solution on YOUR website!
The probability that the new drug will be approved by the FDA is **0.86** (or **86%**).
This problem uses the **Law of Total Probability** to combine the probabilities of approval under two different scenarios: with side effects and without side effects.
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## 🎲 Probability Calculation
Let $A$ be the event that the drug is **Approved**.
Let $S$ be the event that the drug causes **Side Effects**.
Let $S'$ be the event that the drug causes **No Side Effects**.
### 1. Identify Given Probabilities
* **Prior Probabilities (Likelihood of Side Effects):**
* $P(S)$ (Probability of Side Effects) $= 0.20$
* $P(S')$ (Probability of No Side Effects) $= 1 - P(S) = 1 - 0.20 = 0.80$
* **Conditional Probabilities (Approval Rates):**
* $P(A|S')$ (Approved **given** No Side Effects) $= 0.95$
* $P(A|S)$ (Approved **given** Side Effects) $= 0.50$
### 2. Apply the Law of Total Probability
The total probability of the drug being approved, $P(A)$, is the sum of the probabilities of being approved in the two distinct scenarios:
$$P(A) = P(A \text{ and } S') + P(A \text{ and } S)$$
$$P(A) = [P(A|S') \cdot P(S')] + [P(A|S) \cdot P(S)]$$
### 3. Substitute and Solve
$$P(A) = (0.95 \cdot 0.80) + (0.50 \cdot 0.20)$$
$$P(A) = 0.76 + 0.10$$
$$P(A) = \mathbf{0.86}$$
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