Problem #$1($problem #$2$ in chapter $16)18$ ptsManagement of the New Fangled Softdrink Company believes that the probability of a customer purchasingRed Pop or the companys major competition, Super Cola, is based on the customers most recent purchase.Suppose that the following transition probabilities are appropriate:TOFrom Red Pop Super ColaRed Pop $0\mathrm{.}90\mathrm{.}10\mathrm{.}1$ Super Cola $0\mathrm{.}9($a$)$ Show the two$-$period tree diagram for a customer who last purchased Red Pop. What is the probabilitythat this customer purchases Red Pop on the second purchase? $($Embed the image in your document!$)($b$)$ What is the long$-$run market share for each of these two products?$($c$)$ A Red Pop advertising campaign is being planned to increase the probability of attracting Super Colacustomers. Management believes that the new campaign will increase to $0\mathrm{.}15$ the probability of acustomer switching from Super Cola to Red Pop. What is the projected effect of the advertisingcampaign on the market shares?Problem #$2($problem #$8$ in chapter $16)16$ ptsThe purchase patterns for two brands of toothpaste can be expressed as a Markov process with the followingtransition probabilities:ToFrom Special B MDASpecial B $0\mathrm{.}900\mathrm{.}10$MDA $0\mathrm{.}050\mathrm{.}95($a$)$ Which brand appears to have the most loyal customers? Explain.$($b$)$ What are the projected market shares for the two brands?Problem #$3($problem #$9$ in chapter $6)6$ ptsSuppose that in the above problem $($Problem #$8$ in chapter $16)$ a new toothpaste brand enters the market suchthat the following transition probabilities exist:ToFrom Special B $($SB$)$ MDA T$-$White $($TW$)$Special B $($SB$)0\mathrm{.}800\mathrm{.}100\mathrm{.}10$MDA $0\mathrm{.}050\mathrm{.}750\mathrm{.}20$T$-$White $($TW$)0\mathrm{.}400\mathrm{.}300\mathrm{.}30$The solution of the system of equations yields the steady state probabilities:SB $=0\mathrm{.}442,$ MDA $=0\mathrm{.}385$ and TW$=0\mathrm{.}173.$Compare these probabilities to your answer for problem #$2$ and determine which brand will suffer mostfrom the introduction of the new brand of toothpaste? ExplainEXTRA CREDIT option $(4$ points$)(3$ points$)$ Set up the system of equation for determining the steady$-$state probabilities for this three$-$state Markov process $($i$.$e$.$ long run market share for the $3$ brands$).(1$ point$)$: solve the system of equations that you set up in part $($a$).$To earn credit, you need to show your work! $($the solution is given in Problem #$3$: $0\mathrm{.}442,0\mathrm{.}385,0\mathrm{.}173)$