Final answer:
The question involves calculating probabilities using the binomial distribution formula, plotting the probability distribution function, and determining cumulative probabilities. The mean and standard deviation are also calculated with given formulas. However, the scenario with dice might not perfectly fit the binomial framework.
Step-by-step explanation:
The question pertains to the concept of binomial distribution in the context of probability and statistics. To calculate the probability (p) of success in a single roll of three dice using the binomial formula, we would first need to define what constitutes a 'success.' However, the scenario with dice does not strictly fit the binomial framework, since a binomial experiment requires only two outcomes (success or failure), whereas a dice roll has multiple outcomes.
If we hypothetically fit this into a binomial scenario, we would define succeess, set the number of trials (n), and the probability of success on a single trial (p), and use these to plot the probability distribution function. We would use the formulas for mean (μ = np) and standard deviation (σ = √npq), where q = 1 - p, to compute the required statistics.
To generate the plot for part a and the histogram in part b, software or a calculator would typically be used. To calculate the mean and standard deviation (part c), the previously mentioned formulas would be applied. The cumulative probability function (part d) can be derived by summing the probabilities of each number of successes from the least up to a given number of successes.
In order to approximate binomial distribution using normal distribution, the conditions np > 5 and nq > 5 must be met. By adding or subtracting 0.5 to the value of x, we achieve a better approximation.