返回题库

基础骰子游戏 II

Basic Dice Game II

专题
Probability / 概率
难度
L4

题目详情

赌场提供一个六面骰游戏：你最多可以掷 3 次。

  • 第一次掷完可选择停并以第一次点数兑现；或继续。
  • 第二次掷完可选择停并以第二次点数兑现；或继续。
  • 若到第三次则必须以第三次点数兑现。

问：该游戏的公平价值是多少？

A casino offers you a game with a six-sided die where you are paid the value of the roll. The casino lets you roll the first time. If you are happy with your roll, you can cash out. Else, you can choose to roll a second time. If you are happy with your roll, you can cash out on this second value. Else, you can choose to roll for your third and final time and cash out on this third value. What is the fair value of this game?

解析

第三次的期望为 3.5。

因此第二次若点数 ≥4\ge 4 则停，否则继续；由此第二次的最优期望为 4.25。

第一次若点数 ≥5\ge 5 则停，否则继续；由此从开始的期望为

143≈4.67.\frac{14}{3}\approx 4.67.

Original Explanation

The fair value of the last roll is 3.5, and thus you should only opt to roll a third time if your second roll is less than 3.5. Your expected value for the second roll is then 36×5+36×3.5=4.25\frac{3}{6} \times 5 + \frac{3}{6} \times 3.5 = 4.25. Similarly, you should only opt to roll your second time if the first roll is less than 4.25. Your expected value for the first roll is thus 26×5.5+46×4.25≈4.67\frac{2}{6} \times 5.5 + \frac{4}{6} \times 4.25 \approx 4.67.