Ad Evaluation Raters

Solution & Explanation

Question 1: If 100 raters each independently evaluate one ad, how many ads are expected to be rated as good?

Explanation:

• We have two types of raters: Careful Raters and Lazy Raters.
• Careful Raters (80% of the group):
  • Probability of rating an ad as good, $P(G|C) = 0.6$
  • Probability of rating an ad as bad, $P(B|C) = 0.4$
• Lazy Raters (20% of the group):
  • Probability of rating an ad as good, $P(G|L) = 1$
  • Probability of rating an ad as bad, $P(B|L) = 0$

Using the law of total probability, the probability that an ad is rated as good is: $P(G) = P(C) \times P(G|C) + P(L) \times P(G|L) = 0.8 \times 0.6 + 0.2 \times 1 = 0.48 + 0.2 = 0.68$

With 100 raters, the expected number of ads rated as good is: $E(G) = 100 \times P(G) = 100 \times 0.68 = 68$

Thus, we expect 68 ads to be rated as good.

Question 2: If one rater evaluates 100 ads, how many ads are expected to be rated as good?

Explanation:

• This scenario is essentially the same as the first, where we are interested in the expected number of good ratings over 100 trials.
• The probability of a single ad being rated as good remains $P(G) = 0.68$.

Therefore, the expected number of ads rated as good when one rater evaluates 100 ads is: $E(G) = 100 \times 0.68 = 68$

Thus, 68 ads are expected to be rated as good.

Question 3: If an ad is rated bad, what is the probability that the rater was lazy?

Explanation:

• Lazy raters always rate an ad as good, so they can never rate an ad as bad.
• Therefore, if an ad is rated as bad, it must have been rated by a careful rater.

Using Bayes' theorem, the probability that a rater was lazy given that an ad is rated bad is: $P(L|B) = \frac{P(B|L) \times P(L)}{P(B)}$ Since $P(B|L) = 0$ (lazy raters never rate ads as bad), the probability is: $P(L|B) = \frac{0 \times 0.2}{P(B)} = 0$

Thus, the probability that the rater was lazy is 0.