(a) Suppose that Martian dice are regular tetrahedra with vertices labeled 1 to \(4 .\) Two such dice are tossed and the sum of the numbers showing is even. Let \(x\) be this sum. Set up the sample space for \(x\) and the associated probabilities. (b) Find \(E(x)\) and \(\sigma_{x}\). (c) Write the exact binomial expression for the probability of exactly fifteen \(2^{\prime}\) s in 48 tosses of a Martian die. Evaluate it by calculator. (d) Evaluate (c) using the normal approximation. (e) Evaluate (c) using the Poisson distribution.

In probability, the sample space is the set of all possible outcomes of an experiment. For the Martian dice problem, each die has 4 faces, labeled from 1 to 4. When two such dice are tossed, we can calculate all possible outcomes by pairing each number from one die with each number from the other die.
This results in a total of 16 possible outcomes since there are 4 faces on each die (4x4 = 16).
To find the sums that are even, we add the numbers on both dice together and check if the sum is even (e.g., 1 + 1 = 2, 1 + 3 = 4).
Let's list all even sums: 2, 4, 6, 8, each with their associated probabilities.
The probability for each even sum is the number of ways that specific sum can occur divided by the total number of outcomes (16 in this case).

The expected value, often denoted as E(x), represents the average outcome if an experiment is repeated many times. For the Martian dice, we calculate the expected value using the formula:
\[ E(x) = \sum (x_i * P(x_i)) \]
where \( x_i \) are the even sums, and \( P(x_i) \) are their associated probabilities.
Multiply each even sum by its probability and then sum up all these products to get the expected value of x.

The binomial distribution represents the number of successes in a fixed number of independent trials, each with the same probability of success. For Martian dice, we want the probability of getting exactly fifteen 2's in 48 tosses.
This is found using the binomial probability formula:
\[ P(X = 15) = \binom{48}{15} (p^{15}) ((1 - p)^{33}) \]
Here, \( p \) is the probability of rolling a 2 (1/4) on any single toss. Evaluate this using the binomial coefficient and basic arithmetic.

The standard deviation measures the amount of variation or dispersion of a set of values. For the Martian dice problem, we use the formula:
\[ \sigma_{x} = \sqrt{ \sum [(x_i - E(x))^2 \, P(x_i)]} \]
Calculate the squared differences between each even sum \( x_i \) and the expected value, multiply each result by the corresponding probability, sum them, and finally take the square root of this sum.

The Poisson distribution is used to model the number of times an event occurs within a fixed interval of time or space. It’s suitable for rare events. For the problem of getting exactly fifteen 2's in 48 tosses, where \( \lambda = np \) (with \( n = 48 \) and \( p = 1/4 \), hence \( \lambda = 12 \)). Use the formula:
\[ P(X = 15) = \frac{\lambda^{15} e^{-\lambda}}{15!} \]
to calculate this probability.

The normal approximation is often used when dealing with a large number of trials in a binomial distribution. This approach approximates the binomial distribution with a normal distribution. Calculate the mean as \( \mu = np \) and variance as \( \sigma^2 = np(1 - p) \). For the Martian dice problem with n = 48 and p = 1/4:
\[ \mu = 48/4 = 12 \]
\[ \sigma^2 = 48 \cdot 1/4 \cdot 3/4 = 9 \]
The standard deviation \( \sigma = 3 \).
Use the z-score formula and normal distribution tables to find the probability.