**Question **# (4) Glenda and John often have to waitfor the bentos after the orders are placed.During the wait, they usually wander inthe neighbourhood. Next to the ramenshop is a community center. Outsidethe community center is a notice boardwhere community news are posted. Oneday while waiting, they notice a call forvolunteers to deliver letter lunches to theelderly. It says:"Letter lunch is anactivity which delivers letters and luncheswith health exercises, to elderly who livealone or who do not have much opportunityto go out due to the pandemic." Each bento box is prepared witha variety of food, eg., eggplant andpepper meat miso, new potato mustardmayonnaise salad, curry-flavoured boiledeggs, etc.. The community center wantsto promote a healthy diet for the elderly, so the number (N) of meat-based items is limited to no more than two in a bento. When the lunch is finished, the elderly would call the organiser to pick the boxes up, and place their next orders. Along with their next orders, the community center also encourages the elderly to return a note. A note either contains a praise of how delicious the food is (S = 2) or no mention of the food (S = 1). Sometimes, no note is found (S 0). Over time, the community center established the following information: P(S = 0|N = 0) = 0, P(S = 1|N = 0) = 0.3; P(S = 0|N = 1) = 0.3,P(S = 1|N = 1) = 0.3; P(S = 0|N = 2) = 0.5, P(S = 1|N = 2) = 0.4. For the sake of organisation and fairness to all, on any given day, the same type of bentos are made. The menu rotates so that over time, N = 0 in 50% of the days, N = 1 in 30% of the days and N = 2 for the remaining days.= (a) Based on the given information, draw a probability tree and use it to construct thejoint PDF between S and N. (b) An elderly returns a bento with a note praising the food. By using the following methods: (i) probability tree (ii) joint PDF table (iii) Bayes Theorem, find the probability that the bento had one meat item. (c) Two elderly who live in different parts of the neighbourhood order these bentos everyday. Let S₁ and S₂ be defined as S, for the the two elderly. Are S₁, S₂ independent?Justify. (d) Suppose S₁, S2 are independent given N, i.e., this situation is called conditional independence. Find the probability that the bento had 1 meat item if S₁ = S₂ = 2. \mathrm{P}\left(S_{1}, S_{2} \mid N\right)=\mathrm{P}\left(S_{1} \mid N\right) \mathrm{P}\left(S_{2} \mid N\right)