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I was practicing my probability and counting solving ability when I came across this rather interesting question on the web:

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 (5–6 pages). Data on recent reviews indicates that 60% of them are short, 30% are medium, and the other 10% are long. Reviews are submitted in either Word or LaTeX. For short reviews, 80% are in Word, whereas 50% of medium reviews are in Word and 30% of long reviews are in Word. Suppose a recent review is randomly selected.
(a) What is the probability that the selected review was submitted in Word format?
(b)f the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?

So for (a), it seems rather simple that it is (.80)(.5)(.3). For (b), I don't know. First am I thinking of (a) correctly and for b, how would you approach it?

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1 Answer

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Actually a) is the harder part I think,

$P(S)=0.6\quad P(M)=0.3 \quad P(S)=0.1\\ P(W|S)=0.8\quad P(W|M)=0.5\quad P(W|L)=0.3\\ \therefore P(W)=0.8\times0.6+0.5\times0.3+0.3\times0.1$

For b)

$P(S|W)=\frac{P(W|S)P(S)}{P(W)}$

of which all the elements can be found from a)

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