Suppose that we roll ten standard six-sided dice. What is...

Name: Problem 15, Chapter 1: Probability and Computing
Uploaded: 2026-07-02T19:00:52-08:00
Duration: 1 min 41 s
Channel: Nick Johnson
Description: Suppose that we roll ten standard six-sided dice. What is the probability that their sum will be divisible by 6 , assuming that the rolls are independent? (Hint; Use the principle of deferred decisions, and consider the situation after rolling all but one of the dice.)

Suppose that we roll ten standard six-sided dice. What is the probability that their sum will be divisible by 6 , assuming that the rolls are independent? (Hint; Use the principle of deferred decisions, and consider the situation after rolling all but one of the dice.)

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Key Concepts

Independence of Random Variables

Independence means that the outcome of one random event does not influence the outcome of another. When dealing with multiple dice rolls, recognizing that each roll is independent allows the probability of combined events to be expressed as the product of their individual probabilities.

Uniform Probability Distribution

A uniform probability distribution means that each outcome in the sample space is equally likely. In the context of rolling a standard dice, every face (or residue class when considered modulo a number) has an equal probability of occurring, which is a key aspect when calculating probabilities of various events.

Modular Arithmetic

Modular arithmetic involves performing calculations with remainders after division by a fixed number. In probability problems, it is often used to analyze divisibility and congruence conditions, such as determining whether a sum of numbers is divisible by a given integer.

Deferred Decisions

The principle of deferred decisions is a method in probability where some decisions are postponed to simplify analysis. By considering the outcome of one event after others have been realized, one can reduce complex, multi-step problems into simpler conditional probabilities.

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