Using the 7-sided die from Problem 4, create a Monte Carlo model in Excel to simulate the total number of dots obtained when throwing 1, 2, 5, 10, and 20 dice respectively. Use 1000 samples and produce a properly labelled histogram for each case. Make observations and conclusions. Submit your Excel file with your assignment. Q4 A random variable is an uncertain variable where there is a number associated with each outcome of the variable. A good example of this is the 6-sided die that is used in games of chance: each throw of the die produces a number from 1 to 6. Consider now a fair die with 7 faces, i.e. it produces 1 to 7 dots on each throw. If such a die is rolled, what is the chance of obtaining a) 6 dots b) 3 dots or 4 dots or 5 dots c) less than 5 dots d) more than 4 dots If two such dice are rolled, what is the chance of obtaining a total of e) 14 dots f) 8 dots g) 3 dots h) 3 dots or 5 dots i) less than 12 dots

Probability Calculations for a Fair 7-Sided Die Each face (1 to 7) has an equal probability: P(X=k) =1 k=1,234,56"7 (a) Probability of rolling a 6 P(X =6) =} ~0.1429 (b) Probability of rolling a 3, 4, or 5 P(X=3ordor5)=1+1+1=3~04286 (c) Probability of rolling less than 5 dots (i.e., 1, 2, 3, or 4) P(X 4) = $ ~0.4286 Two Dice Rolls: Probability of Different Sums Since each die i independent, there are 49 possible outcomes (7 x 7). (e) Probability of obtaining a total of 14 Only one way: (7.7) P(S = 14) = 4 ~0.0204 (f) Probability of obtaining a total of 8 Possible sums: (1,7), (2,6), (3,5), (4.4), (5,3), (6,2), (7.1) = 7 ways P(S=8) =1 =1~01429 (g) Probability of obtaining a total of 3 Possible sums: (1,2), (2,1) - 2 ways P(S =3) = % ~0.0408 (h) Probability of obtaining a total of 3 or5 *S=3-(12), (21 - 2ways *S=5-14),(23), (32, 41> 4 ways * Total ways: 6 P(S=3o0r5)=f; ~01224 (i) Probability of obtaining a total less than 12 Adding probabilities for sums 2 to 11: Sum Ways 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 6 10 5 n 4 Total ways: 43 P(S