Question 1166559
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

$$P(A) = (0.95 \cdot 0.80) + (0.50 \cdot 0.20)$$

$$P(A) = 0.76 + 0.10$$

$$P(A) = \mathbf{0.86}$$