Explain the following question how to do using R software code with output (screenshot)

PLEASE

1. At a school fete fundraiser, an unbiased spinning wheel has numbers 1 to 50 inclusive.

a) Find the probability of getting a multiple of 7 in one spin of the wheel.

b) If the wheel is spin 500 times during the day, what is the likelihood of getting a multiple of 7 more than 15% of the time?

c) Suppose 20 people play each time the wheel is spin. When a multiple of 7 comes up, $5 is paid to players, but when it does not the players must pay $1.

i) How much would the wheel be expected to make or lose for the school if it was spin 500 times?

ii) Find the probability that the school will lose money if the wheel is spin 500 times during the day.

2. When typing a document, trainee typists make on average 2.5 mistakes per page. The mistakes on any one page are made independently of any other page. Suppose X represents the number of mistakes made on one page, and Y represents the number of mistakes made in a 52-page document.

a) State the distributions of X and Y . You may assume that the sum of n independent Poisson random variables, each with mean m, is itself a Poisson random variable with mean mn.

b) Rana is a trainee typist. Find the probability that Rana will make:

(i) more than 2 mistakes on a randomly chosen page

(ii) more than 104 mistakes in a 52-page document.

c) Now assume that X and Y can be approximated by normal random variables with the same means and variances as found above. Use the normal approximations to estimate the probabilities in b.

3. Adam does the crossword in the local newspaper every day. The time taken by Adam, X minutes, to complete the crossword is modelled by the normal distribution N(22, 5.2 ).

(a) Given that, on a randomly chosen day, the probability that he completes the crossword in less than a minutes is equal to 0.8, find the value of a.

(b) Find the probability that the total time taken for him to complete five randomly chosen crosswords exceeds 120 minutes.

Beatrice also does the crossword in the local newspaper every day. The time taken by Beatrice, Y minutes, to complete the crossword is modelled by the normal distribution N(40, 62 ).

(c) Find the probability that, on a randomly chosen day, the time taken by Beatrice to complete the crossword is more than twice the time taken by Adam to complete the crossword. Assume that these two times are independent.