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To find the probability that a rater is lazy given that they rated three ads as good, we use Bayes' Theorem:
P(L∣G3)=P(G3)P(G3∣L)⋅P(L)
Where:
The total probability of all ads being rated as good, P(G3), is:
P(G3)=P(L)P(G3∣L)+P(Lc)P(G3∣Lc)=0.2⋅1+0.8⋅0.216=0.3728
Thus, the probability that a rater is lazy given they rated all three ads as good is:
P(L∣G3)=0.37280.2⋅1=0.5365
Explanation: This means there's a 53.65% chance the rater is lazy if they rate all three ads as good.
We need to determine how the probability of a rater being lazy changes as they rate all N ads as good, where N approaches infinity:
P(L∣GN)=P(GN)P(L)P(GN∣L)
Where:
The total probability P(GN) is:
P(GN)=P(L)P(GN∣L)+P(Lc)P(GN∣Lc)=0.2+0.8⋅0.6N
As N→∞, 0.6N→0, so:
P(GN)→0.2
Thus:
P(L∣GN)=0.20.2=1
Explanation: As N increases, the probability that a rater is lazy approaches 100% if they rate all ads as good.
To filter out lazy raters, we can set a threshold α for the probability P(L∣GN). We flag a rater as lazy if:
P(L∣GN)>α
Rewriting:
0.2+0.8⋅0.6N0.2>α
Solving for N, we get:
0.2>α(0.2+0.8⋅0.6N)
0.2(1−α)>0.8⋅0.6N⋅α
Taking the natural logarithm:
ln(0.2)+ln(1−α)<ln(0.8)+N⋅ln(0.6)+ln(α)
Solving for N:
N=⌈ln(0.6)ln(0.2)+ln(1−α)−ln(0.8)−ln(α)⌉
Explanation: This formula provides the minimum number of consecutive good ratings required to flag a rater as lazy with confidence level 1−α. For example, with α=0.05, we find that a rater needs to rate at least 9 ads as good to be flagged as lazy with 95% confidence.