Learning From Data – A Short Course: Exercise 5.2

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Suppose that for 5 weeks in a row, a letter arrives in the mail that predicts the outcome of the upcoming Monday night football game. You keenly watch each Monday and to your surprise, the prediction is correct each time. On the day after the fifth game, a letter arrives, stating that if you wish to see next week’s prediction, a payment of 50.00 USD is required. Should you pay?

(a) How many possible predictions of win-lose are there for 5 games?

2^{5} = 32.

(b) If the sender wants to make sure that at least one person receives correct predictions on all 5 games from him, how many people should he target to begin with?

32. A half receives win prediction and the other half receives lose prediction.

(c) After the first letter ‘predicting’ the outcome of the first game, how many of the original recipients does he target with the second letter?

16 of 32. A half receives win prediction and the other half receives lose prediction.

(d) How many letters altogether wil have been sent at the end of the 5 weeks?

2^{5} + 2^{4} + 2^{3} + 2^{2} + 2^{1} + 2^{0} = 63.

(e) If the cost of printing and mailing out each letter is 0.50 USD, how much would the sender make if the recipient of 5 correct predictions sent in the 50.00 USD?

50.00 - 63 \times 0.50 = 18.5.

(f) Can you relate this situation to the growth function and the credibility of fitting the data?

Check out the Lecture 17 for more about this exercise. In short, the results of 5 games are 5 data points. The 5 predictions of 5 games sent to one person represents a dichotomy. If the hypothesis set can shatter data set of size 5 (with binary output), then it’s obvious and meaningless to have a hypothesis in such hypothesis set fit the given 5 data points.


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