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Numerade Educator

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Problem 54 Medium Difficulty

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.
Consider randomly selecting such a coffee purchaser.
(a) What is the probability that the individual purchased a small cup? A cup of decaf coffee?
(b) If we learn that the selected individual purchased a small cup, what now is the probability that s/he chose decaf coffee, and how would you interpret this probability?
(c) If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability from (a)?

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Video Transcript

all right. For the first part of the problem, we want to figure out what the probability of a small cup is and what the probability of decaf is. I'll use the letters that I have up table soapy of p of S and P of B. The probability of the getting a small is just going to be the some of the two row elements for a small. So we have 14% less 20% which is going to give us a total of 34%. Or we can write that as 0.34 and similarly, the probability of decaf, which is going to be 20% plus 10% plus 10% it's going to be 40% or 0.4 for be. We've learned that some if we know somebody that are excuse me. If we know that somebody has purchased a small cop, we want Thio. Then determine what the probability is that they have purchased a decaf and we want to interpret this probability so we'll start with the interpretation first. So we know we know small than we want probability decaf. So we can interpret that as meaning we want to find the probability of decaf given small. We know that event. The event where they have purchased a small coffee occurred. We want to know if it also occurred that they purchased a decaf. So the probability of D given s is the probability of the intersection s divided by the probability of s So the probability of D intersection s you could just read directly off from the table here. Uh, the intersection of D. N s is 20%. So we have 0.2 divided by the probability of getting a small is 34 or 0.34 This comes out to about 0.588 Or we can approximate that to 0.59 Uh huh. Mhm for C. I want to go in the other direction. We know decaf. We want probability small so we can interpret that as the opposite of this p of s given D, which is again the probability of the Intersect s on top divided by now. The probability of d you know that the probability of the Intersect s as we saw it before 0.2 probability of D is 0.4. So that gives us that the probability of small given decaf is 0.5. We also wants Thio comment on how this compares to the corresponding unconditional probability from part A. So if we know that somebody's purchased a decaf coffee, it is more likely that they have gotten a small than in the general case where we don't know what kind of coffee they've gotten.