1. **State the problem:** We need to find which probability distribution table has a mean (expected value) of $7$ for the random variable $X$ representing profit. 2. **Recall the formula for the mean of a discrete probability distribution:** $$\mu = E(X) = \sum (x \cdot P(x))$$ where $x$ are the values of the random variable and $P(x)$ are their probabilities. 3. **Check each table by calculating the mean:** - Top-left table: $$5 \times 0.4 + 8 \times 0.5 + 10 \times 0.1 = 2 + 4 + 1 = 7$$ - Top-right table: $$5 \times 0.2 + 8 \times 0.7 + 10 \times 0.1 = 1 + 5.6 + 1 = 7.6$$ - Bottom-left table: $$5 \times 0.3 + 8 \times 0.5 + 10 \times 0.2 = 1.5 + 4 + 2 = 7.5$$ - Bottom-right table: $$5 \times 0.6 + 8 \times 0.1 + 10 \times 0.3 = 3 + 0.8 + 3 = 6.8$$ 4. **Interpretation:** Only the top-left table has a mean of exactly $7$. **Final answer:** The top-left table represents the probability distribution for $X$ with mean $7$.