But the answer is 7/9

If two-thirds of patients survive their first MI, then the proportion who die from it is:

• The proportion of patients who survive the first MI is $\frac{2}{3}$.
• Of these survivors, $\frac{1}{3}$ live 10 years. This means the proportion of these survivors who die within 10 years is $1 - \frac{1}{3} = \frac{2}{3}$.
• To find the proportion of all patients who survive the first MI and then die within 10 years, multiply these probabilities: $$P(\text{survive MI and die within 10 years}) = P(\text{survive MI}) \times P(\text{die within 10 years | survive MI})$$ $$P(\text{survive MI and die within 10 years}) = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$$

This total is the sum of those who die from the first MI (Step 1) and those who survive the first MI but die within 10 years (Step 2).

To add these fractions, find a common denominator, which is 9:

Your answer of $\frac{7}{9}$ is correct!