Probability Theory | Coin Tossing Problems (TBC) | 概率论中的抛硬币问题 （未完待续）

Q1

Question:

Suppose we play a game. I roll a die up to three times. Each time I roll, you can either take the number showing as dollars, of roll again. What is your expected winnings?

Solution:

Let random variable $X_k$ be the winnings for rolling a die up to k times.
The key is to formulate the following strategy: after rolling a dice, if the number showing is greater than the expected winnings of rollings the rest of the times, you should take the money. Expressing this with math:

$E[X_k]=E[\max$ (number for this toss, $E[X_{k-1}])$

Thus, working from the end:
$$E[X_1]=1/6\times \sum _{i=1}^{6}i=7/2$$
$$E[X_2]=1/6\times (6+5+4)+1/2\times E[X_1]=17/4$$
$$E[X_3]=1/6\times (6+5)+2/3\times E[X_2]=14/3$$

Reflection:

This kind of problem is a little bit like DP - defining a subproblem, finding the relationship between subproblems, then starting from a base case and working upwards step by step.