Just a quick Q on the last question (c). Based on research of the average lifetime a couple assumes a probability of $0.75$ that the husband will still be alive in $20$ years while the wife has a chance of $0.8$. How likely are the following events?
(a) Both are still alive in $20$ years.
$$P(A∩B)= 0.75 \cdot 0.8$$
(b) None of them is still alive in $20$ years.
$$P(A∩B)= 0.25 \cdot 0.2$$
(c) At least one of them is still alive in $20$ years.
Now for c I used
$$P(A∪B)= 0.75 + 0.8 -(0.75\cdot0.8)$$
but my friend said it's wrong and said I should use $(A∩B)$ but with additions. Is this correct? Was I wrong?
You're correct and your friend is incorrect.
Alternatively for $(c)$ you could have done
$$P(A\cup B)=(0.75\cdot 0.2)+(0.25\cdot 0.8)+(0.75\cdot 0.8)=0.95$$
which agrees with your solution.
One of three things can happen to satisfy $A\cup B$
$A$ happens and $B$ doesn't happen
$A$ does not happen and $B$ happens