1. **State the problem:** We have 30 cupcakes in total. - 16 cupcakes are iced. - 3 cupcakes are both iced and contain walnuts. - 5 cupcakes are neither iced nor contain walnuts. We want to find the probability that a randomly picked cupcake contains walnuts. 2. **Draw and label the Venn diagram:** - Let the circle for iced cupcakes have 16 total. - The intersection (iced and walnuts) is 3. - Let the number of cupcakes containing walnuts but not iced be $x$. - Outside both circles are 5 cupcakes. 3. **Calculate the number of cupcakes containing walnuts:** - Total cupcakes = 30 - Cupcakes neither iced nor walnuts = 5 - Cupcakes iced only = $16 - 3 = 13$ 4. **Find $x$ (cupcakes containing walnuts but not iced):** Total cupcakes = iced only + walnuts only + both + neither $$30 = 13 + x + 3 + 5$$ Simplify: $$30 = 21 + x$$ Subtract 21 from both sides: $$30 - 21 = \cancel{21} + x - 21$$ $$9 = x$$ 5. **Calculate total cupcakes containing walnuts:** $$\text{Walnuts} = x + 3 = 9 + 3 = 12$$ 6. **Calculate the probability:** $$P(\text{walnuts}) = \frac{\text{number of cupcakes with walnuts}}{\text{total cupcakes}} = \frac{12}{30}$$ Simplify the fraction: $$\frac{12}{30} = \frac{\cancel{6} \times 2}{\cancel{6} \times 5} = \frac{2}{5}$$ **Final answer:** The probability that a randomly picked cupcake contains walnuts is $\boxed{\frac{2}{5}}$.