Interactive Dice Probability Explorer

Problem 1.4.7: Dice Until Target Sums

Part (a): A pair of dice is cast until either the sum of seven or eight appears. Show that the probability of a seven before an eight is 6/11.

Part (b): Next, this pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show the probability of the six and eight occurring before two sevens.

Customize the Problem

Target Sum 1 (Part a): Target Sum 2 (Part a):

Target Sum A (Part b): Target Sum B (Part b): Losing Sum (Part b): Losing Count:

Sum: -

Game Mode: Part A Part B

Part (a): Probability Analysis

Step 1: All Possible Dice Outcomes

Method 1: Direct Probability

Method 2: Geometric Series

Step-by-Step Calculation

Part (b): Markov Chain Analysis

Step 1: State Definition

We define states as: (Number of losing sums seen, Target A seen?, Target B seen?)

Terminal states: WIN = both targets seen before losing limit, LOSE = losing limit reached