Nate (similar to Freddy, who was discussed in class 19) is trying to decide whether Kimaya is a nice person. There is an equal chance that Kimaya is mean, average, or nice. Whenever a person meets Kimaya, she acts either pleasantly or unpleasantly. If Kimaya is mean, she will act pleasantly one-quarter of the time; if she is average, she will act pleasantly one-half of the time; and if she is nice, she will act pleasantly three-quarters of the time (everyone has their bad days). In reality, Kimaya’s behavior is independent over time. But Nate believes in the law of small numbers, and has N = 24 (That is to say, Nate believes that he is drawing from a set of 24, without replacement). (a) Suppose Nate meets Kimaya four times, and she is pleasant each time. This part asks you to calculate the correct conclusions Nate should draw from this. i. If Kimaya is mean, calculate the true probability that Kimaya is pleasant four times in a row.
• Hint:
If they meet only once, what is the true probability that a mean Kimaya is pleasant that time? If they meet twice, what is the true probability that a mean Kimaya is pleasant both times? If they meet three times, what is the true probability that she is pleasant all three times?
ii. If Kimaya is average, calculate the true probability that Kimaya is pleasant four times in a row.
iii. If Kimaya is nice, calculate the true probability that Kimaya is pleasant four times in a row.
iv. Based on the answers to the previous parts, calculate the overall probability that Kimaya is pleasant four times in a row.
v. Apply Bayes’ rule and calculate the probability that Kimaya is nice conditional on the fact that she was observed to be pleasant four times in a row. That is to say, if Nate has seen Kimaya be pleasant all four times, what is the probability that she is truly nice?
(b) This part asks you to calculate the conclusions Nate draws from seeing Kimaya be pleasant four times.
i. Calculate the probability with which Nate thinks Kimaya is pleasant four times in a row if she is mean.
ii. Calculate the probability with which Nate thinks Kimaya is pleasant four times in a row if she is average.
iii. Calculate the probability with which Nate thinks Kimaya is pleasant four times in a row if she is nice.
iv. Based on the answers to the previous parts, calculate the overall probability that Nate thinks Kimaya is pleasant four times in a row.
v. Apply Bayes’ rule and calculate the probability with which Nate thinks Kimaya is nice conditional on the fact that she was pleasant four times in a row.
(c) Explain the general intuition behind the difference in your final answers to parts (a) and (b).
(d) Nate decides that he is not yet sure whether Kimaya is nice, so that he should meet Kimaya regularly to find out. After a while, he meets her nineteen more times. Before the first of these nineteen meetings takes place, the balls in his imaginary urn are replenished (and they are not replenished again for the duration of these nineteen meetings). This time Kimaya is pleasant twelve times and not pleasant seven times. Does this increase or decrease Nate’s confidence that Kimaya is nice? Should it? Explain the intuition. [Hint: Although you can answer this question using Bayes’ rule, there is a simpler way to do it. Namely, you can ask whether a mean, average, or nice Kimaya is most likely to be pleasant twelve times and not pleasant seven times, and what Nate thinks about these likelihoods.]