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The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short $(1-2$ pages), medium $(3-4$ pages) or long $(5-66$ pages. Data on recent reviews indicate that 60$\%$ of them are short, 30$\%$ are medium, and the other 10$\%$ are long. Reviews are submitted in either $\mathrm{Word}$ or LaTeX. For short reviews, 80$\%$ are in $\mathrm{Word}$ , whereas 50$\%$ of medium reviews and 30$\%$ of long reviews are in $\mathrm{Word}$ . Suppose a recent review is randomly selected.

(a) What is the probability that the selected review was submitted in Word?

(b) If the selected review was submitted in Word, what are the posterior probabilities of it being short, medium, and long?

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Okay, So in this problem we are discussing the reviews for a certain scientific journal. Um, we consider and the reviews could be classified as short, medium or long. And they could be written with word. Or they may be written with latex, which would be specified as w prime. So 60% of the papers are short, 30% are medium one or a 10% are long. And then of the short papers, 80% are written in word of the medium papers, 50% of Britain inward. And of the long papers, 30% are written in work. The first problem we want Thio solve is we want to find out what the probability that a particular that any review was submitted inward so we can find that out Using the love total probability. So pfw is equal to p of W given s times p of s and I'll give the justification. So each of these options the S M and L are exclusive. A paper can't be short and medium and they were exhaustive. If it's not short, medium or long, then it's not possible. So we can apply total love Total probability. So you have p of w given s time two p of s plus the p of w given em times the probability of em plus the probability of W given l Times the probability of l which we can plug in our values. We have 0.8 times 0.6 plus zero, five times 0.3 plus 0.3 times 0.1 That comes out to 0.66 then. So that's for a from B. I want to answer If a selected review was submitted inward, what are the posterior possibility? Probability is rather of it being short, medium or long. So what that means is we want to find a p. M s Given w p of n Given W and p of l given W. P. M s given w can figure this out using Bayes theorem. That would be p of W given s times p of s divided by p of w give w we figured out before, So that would be 0.8 times 0.6 0.6, divided by 0.66 That comes out to 0.73 p. F. M. Given value is similar deal here just with a M and seven s. So that is 0.5 times 0.3, divided by 0.66 which gives US 0.23 and p. M L Given W is 0.3 times 0.1, divided by 0.66 gives US 0.4