Question 1166561
The probability that the new drug will be approved by the FDA is **0.86**.

This problem is solved using the **Law of Total Probability**. The drug can be approved under two mutually exclusive scenarios: with no side effects or with side effects.

## 📝 Calculation using the Law of Total Probability

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 the Given Probabilities

| Event | Notation | Probability |
| :---: | :---: | :---: |
| Drug causes side effects | $P(S)$ | $0.20$ |
| Drug causes no side effects | $P(S') = 1 - P(S)$ | $1 - 0.20 = 0.80$ |
| Approved *given* no side effects | $P(A|S')$ | $0.95$ |
| Approved *given* side effects | $P(A|S)$ | $0.50$ |

### 2. Apply the Law of Total Probability

The probability of approval, $P(A)$, is the sum of the probabilities of being approved in each scenario:

$$P(A) = P(A|S') \cdot P(S') + P(A|S) \cdot P(S)$$

### 3. Substitute the Values

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

### 4. Compute the Result

$$P(A) = 0.76 + 0.10$$
$$P(A) = \mathbf{0.86}$$

The probability the drug will be approved by the FDA is $0.86$.