Exercise 5 Three dice are tossed, one red, one blue, and one green. a. What is the probability that the sum of the three faces showing equals 7? b. Write an R program that simulates this random experiment 10 times and calculates the proportion of simulations in which the sum of the three faces showing is 7. Include the program and it's output in your homework submission. Hint 1: By the Law of Large Numbers, the number you find should be close to the theoretical probability found in question 1. Hint 2: In your R program, you can use a 'for' loop to repeat the simulation 10 times (each time, simulate the 3 dice roll and store their sum; at the end, find the proportion of 7's out of all results). A faster method is to simulate a dice roll 3x10 times, arrange this vector into a matrix with 3 rows and 10 columns (or the other way around), calculate the column sums of the matrix, and finally determine the proportion of 7's among these sums. Exercise 6 (Textbook 2.146) Five cards are drawn without replacement from a standard 52-card playing deck. What is the probability that all 5 cards will be of the same suit? Note: a standard 52-card playing deck has 4 suits: clubs, diamonds, hearts and spades with 13 cards for each suit. https://en. wikipedia.org/wiki/Standard_52-card_deckExercise 1 To join a certain club, a person must be either a statistician or a mathematician or both. Of the 30 members in this club, 24 are statisticians and 19 are mathematicians. How many persons in the club are both a statistician and a mathematician? Exercise 2 Let A, B, and C be any three events. Use Venn diagrams to show that a. An (BUC) = (An B) U (AnC) b. AU (Bn C) = (A UB) n (A UC)Exercise 3 Patients arriving at a hospital outpatient clinic can select one of four stations for service. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Four patients arrive at the clinic and their selection of stations is observed. a. What is the size of the sample space for the experiment? b. Let A be the event that each station receives a patient. List the sample points in A. c. Make a reasonable assignment of probabilities to the sample points and find P(A). Exercise 4 The Florida Cash 3 daily midday lottery number consists of three digits, each 0-9. (Digits can be repeated: for example, 112. Order matters: 123 is not the same as 213 or 321.) a. How many possible midday numbers are there? b. If all the midday numbers are equally likely, find the probability that exactly two of the digits are the same