For example, if you throw a die, you have two chances to throw. You can choose to throw once or twice, but the number of the last throw will be the final result. Find the mathematical expectation of the points obtained by such a throw. If you have three chances, still taking the number of the last throw as the final result, find the mathematical expectation of the points obtained by such a throw.
Analyzing how many times one should roll is the key point of this problem.
Solution: If only one roll is made, the expected value of the points obtained is: $\frac{1}{6} \times(1+2+3+4+5+6)=3.5$;
If two rolls are made, after the first roll, there is still one more roll opportunity, with an expected value of 3.5; thus, if the first roll is $1,2,3$ points, then re-roll. Therefore, the expected value of rolling twice is: $\frac{1}{6} \times(4+5+6)+\frac{1}{2} \times 3.5=\frac{17}{4}$.

If three rolls are made, after the first roll, there are still two more roll opportunities, with an expected value of $\frac{17}{4}=4.25$, thus if the first roll is 1,2,3,4 points, then re-roll. Therefore, the expected value of rolling three times is: $\frac{1}{6} \times(5+6)+\frac{4}{6} \times \frac{17}{4}=\frac{14}{3}$.
!Comment: How to choose the optimal rolling method according to the rules is the key to this problem.