2. Consecutive Lottery Numbers The California Lottery\'s SuperLottoPlus game inv

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2. Consecutive Lottery Numbers The California Lottery's SuperLottoPlus game involves picking five numbers between 1 and 47 Many people who play the lottery refuse to choose consecutive integers among their five number choices, believing that it's very unlikely for consecutive numbers to appear? But how unlikely is it? We will use simulation to approximate this probability (a) First, make a guess for the probability that consecutive numbers appear Simulation Analysis: (b) Use R to conduct a simulation of the California game by randomly selecting 5 numbers (without replacing) from the numbers 1-47, repeating this process 25 times. R: winner sample (1:47,5) Repeat this process a total of 25 times, (you can simply copy-and-paste the commands above over and over), keeping a running tally of whether or not the (sorted) set includes consecutive integers. Report the proportion of these 25 drawings in which the numbers include at least one set of consecutive integers. (c) Now we want to automatic this process to run it a very large number of times R: >winner0y 0; diff s for (j in 1:1000) ( winner sample (1:47,5) sortwin sort (winner) for (i in 1:4) difrtwinli+11-sortin(4] y((min (diff)-1) table (y) Write a description of what each commands does, and explain how the computer achieves the goal of determining whether a drawing produces at least one set of consecutive integers. (d) From your results in (c), report the estimate of the probability that this lottery game produces at least one set of consecutive integers. How does this compare to your prediction in (a)? Exact Analysis: (e) How many different drawings are possible? In other words, how many ways are there to choose five numbers from 47 numbers? (0 It can be shown that when choosing k numbers between 1 and n, the number of ways to get sis(2)-(n-k +) /. Use this result to determine the number of different consecutive integers is drawings that include consecutive integers with the Califormia game. inte (8) Use your answers to (e) and () to determine the theoretical probability that the five winning numbers in the California game will include consecutive integers. (h) How does the theoretical probability compare to your simulation results in (d)? (i) Even though picking consecutive numbers won't increase your chances of winning in any one drawing, explain another advantage to doing so considering other players' tendencies.

Explanation / Answer

Exact Analysis

e) 47C5 = (47*46*45*44*43)/ (1*2*3*4*5)=1533939

f)n=47, k=5

so 47C5 - 43C5 = (47*46*45*44*43 - 43*42*41*40*39)/(5*4*3*2*1) = 571341

g)P = 571341/1533939 = 0.372

i) As all will be picking numbers which are not consecutive, you will have an advantage when you pick consecutive numbers because when you hit a jackpot you dont have to share it. The occurrence of this consecutive number is as any other combination.

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