MHB Discrete Probability Distribution Tables Skills

[drumsticksss]

A problem i made up for some of my friends who need help with discrete distributions tables. Can you do it?
Dice Generator
Part I:
1. Construct a discrete probability distribution table for a fair six-sided dice. (Round according to example)
2. Calculate the mean, variance, and standard deviation based on the probability distribution.

Part II
A dice simulator was used to “roll” sixty six-sided dice. The results are provided below.
2 4 2 4 3 1
4 3 3 1 5 5
6 2 2 1 1 4
4 4 3 1 5 6
1 2 3 2 5 2
1 4 1 5 1 6
5 4 2 3 2 4
6 4 1 4 5 1
3 6 3 3 4 1
6 6 2 1 2 3

1. Construct a discrete probability distribution table based on the data from the simulator. (Round according to example)

2. Calculate the mean, variance, and standard deviation based on the data.

3. Compare the classical probabilities from Part I with the empirical probabilities from Part II. What are the differences in the probabilities for each possible value? Make a table displaying the differences.
Part Ix p(x) x*p(x) x (x-µ)2 (x-µ)2*p(x)
1 0.1667 0.1667 -2.5007 6.2535 1.042
2
3
4
5
6
∑x*p(x) = ∑(x-µ)2*p(x)=Part II
x p(x) x*p(x) x-µ (x-µ)2 (x-µ)2*p(x)
1 0.2167 0.2167 -2.1671 4.6963 1.018
2
3
4
5
6
∑x*p(x)= ∑(x-µ)2*p(x)=Differences:

x Classical (Part I) Empirical (PartII) Differences
1 0.1667 0.2167 -0.05
2
3
4
5
6

 

[member38644]

∑p(x)= ∑p(x)=

The differences in probabilities between the classical and empirical data are small, with the largest difference being -0.05 for the value of 1. This could be due to the small sample size in Part II compared to the theoretical probabilities calculated in Part I. However, overall, the empirical probabilities are close to the theoretical probabilities, indicating that the dice simulator is producing fairly accurate results.