Question

Two fair dice are rolled. Let $X$ equal the product of the 2 dice. Compute $P\{X=i\}$ for $i=1, \ldots, 36$ 

   Two fair dice are rolled. Let $X$ equal the product of the 2 dice. Compute $P\{X=i\}$ for $i=1, \ldots, 36$
A First Course in Probability
A First Course in Probability
Sheldon Ross 8th Edition
Chapter 4, Problem 2 ↓
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Two fair dice are rolled. Let $X$ equal the product of the 2 dice. Compute $P\{X=i\}$ for $i=1, \ldots, 36$
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Key Concepts

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Probability Mass Function
The probability mass function (PMF) assigns probabilities to the values of a discrete random variable. In this problem, the PMF of the product is determined by the ratio of the number of outcome pairs producing a given product to the total number of outcome pairs (36).
Independence
The concept of independence implies that the outcome of one die does not affect the outcome of the other. This property allows for the multiplication of probabilities when considering the joint outcomes of two dice.
Uniform Sample Space
When rolling two fair dice, every ordered pair of outcomes has an equal probability. This uniformity simplifies probability calculations, as the overall probability is computed by dividing the number of favorable outcomes by the total number of outcomes in the sample space.
Discrete Random Variables
This refers to variables that take on a countable number of distinct values. In the context of rolling dice, the product of the outcomes is a discrete random variable whose value can be determined by counting the number of outcomes that give that product.
Counting Methods and Combinatorics
Computing the probability that a function of outcomes (such as the product) equals a specific value involves counting the number of valid outcome pairs that satisfy this condition. This is a direct application of combinatorial principles in probability.
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Transcript

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00:01 So in the question it was stated that two fair guys ruled, i have to determine the probability that the sum of the two dies is 12.
00:11 So i'll now find out this.
00:14 Let's assume this to this first dice.
00:20 The results of first dice can be 1, 2, 3, 4, 5, and 6.
00:30 The same goes for the second dice...
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