Binomial Probability Intermediate Question

Question

Question: The probability that persons aged 60 and 65 will survive the next year is 0.97 and 0.96 respectively. From a group of five people of whom three are aged 60 and two aged 65, what is the probability that (i) not more than one will die within the next year, (ii) the two aged 65 survive and not more than one dies within the next year?

Not sure how to do part i) My attempt was to consider two cases and add together. This includes when a 60 yr old dies, then when a 65 year old dies. However my answer is wrong compared to answers and I don't know what I did wrong

$$ \binom {3} {3} (0.97)^3*\binom {2}{1}(0.96)(0.04)+\binom {3} {2} (0.97)^2(0.03)*\binom {2}{2}(0.96)^2 $$

Answer 1

So far you have only consider exactly one death.

You have yet to consider the case where there is no death. $(0.97)^3(0.96)^2$.

Answer 2

You are close in your answer to this very grim probability question. It looks like you just missed the case when no one dies, which is just $${{3}\choose{3}} (0.97)^3 {{2}\choose{2}} (0.96)^2$$ If you add this to your answer, do you get the correct one?