Probability Theory | Coin Tossing Problems (TBC) | 概率論中的拋硬幣問題 （未完待續）

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.