1. **State the problem:** We have a discrete random variable $X$ representing the number of patients visiting a clinic each day with probabilities: $$P(0) = 0.2, \quad P(1) = 0.3, \quad P(2) = 0.5$$ We want to find the expected value (mean) of $X$, denoted $E(X)$. 2. **Formula for expected value:** For a discrete random variable, $$E(X) = \sum_{i} x_i P(x_i)$$ where $x_i$ are the possible values and $P(x_i)$ their probabilities. 3. **Apply the formula:** $$E(X) = 0 \times 0.2 + 1 \times 0.3 + 2 \times 0.5$$ 4. **Calculate each term:** $$0 \times 0.2 = 0$$ $$1 \times 0.3 = 0.3$$ $$2 \times 0.5 = 1.0$$ 5. **Sum the terms:** $$E(X) = 0 + 0.3 + 1.0 = 1.3$$ 6. **Interpretation:** The expected number of patient visits per day is $1.3$. This means on average, about 1.3 patients visit the clinic daily.