Suppose that we independently roll two standard six-sided...

Suppose that we independently roll two standard six-sided dice. Let $X_{1}$ be the number that shows on the first die, $X_{2}$ the number on the second die, and $X$ the sum of the numbers on the two dice. (a) What is $\mathbf{E}\left[X \mid X_{1}\right.$ is even $] ?$ (b) What is $\mathbf{E}\left[X \mid X_{1}=X_{2}\right] ?$ (c) What is $\mathbf{E}\left[X_{1} \mid X=9\right]$ ? (d) What is $\mathbf{E}\left[X_{1}-X_{2} \mid X=k\right]$ for $k$ in the range $[2,12] ?$

Suppose that we independently roll two standard six-sided dice. Let $X_{1}$ be the number that shows on the first die, $X_{2}$ the number on the second die, and $X$ the sum of the numbers on the two dice.
(a) What is $\mathbf{E}\left[X \mid X_{1}\right.$ is even $] ?$
(b) What is $\mathbf{E}\left[X \mid X_{1}=X_{2}\right] ?$
(c) What is $\mathbf{E}\left[X_{1} \mid X=9\right]$ ?
(d) What is $\mathbf{E}\left[X_{1}-X_{2} \mid X=k\right]$ for $k$ in the range $[2,12] ?$

Key Concepts

Conditional Expectation

Conditional expectation is the expected value of a random variable given that some condition or event has occurred. It involves recalculating probabilities and outcomes based on known information and is fundamental in problems where the outcome of one event influences or provides information about another.

Independence of Random Variables

Independence means that the outcome of one random variable has no effect on the outcome of another. In the context of dice, each die roll is independent, which simplifies analysis since the probability of a combined event can be expressed as the product of the individual probabilities.

Uniform Distribution in Dice Rolls

A standard die roll follows a uniform distribution, meaning each face (number) has an equal chance of appearance. This uniformity is central to computing expectations because it ensures symmetry across outcomes and simplifies the determination of probabilities for events, particularly when conditioning on a subset of the sample space.

Symmetry in Probability Distributions

Symmetry in probability distributions refers to situations where outcomes are balanced or mirror each other, often leading to simplified calculations. In problems with dice, recognizing symmetry—such as equal likelihood of outcomes when conditioned on events like equal dice values or a fixed sum—can be used to derive expectations without detailed enumeration of every outcome.

Linearity of Expectation

Linearity of expectation states that the expected value of the sum (or difference) of random variables is equal to the sum (or difference) of their expected values, regardless of whether the variables are independent. This property is especially useful in dice problems involving sums or differences and allows for breaking down complex conditional expectations into simpler components.

Transcript

Please give Ace some feedback