Netflix Movie Raters
Netflix has hired people to rate movies. Out of all of the raters, 80% of the raters are "careful" and rate 60% of the movies they watch as "good" and 40% as "bad". The other 20% of raters are "lazy" and rate 100% of the movies they watch as "good".
Assuming all raters rate approximately the same number of unique movies (or a movie is randomly assigned to a rater), what is the overall probability that a movie is rated "good"?
Related Concepts
Hint
To find the overall probability that a movie is rated "good", we need to consider the two types of raters separately and then combine their contributions.
- Define your events:
- Let G = the event that a movie is rated "good".
- Let C = the event that the rater is "careful".
- Let L = the event that the rater is "lazy".
- What are the probabilities of encountering each type of rater?
- P(C) = Probability of a rater being careful = ? (from the problem)
- P(L) = Probability of a rater being lazy = ? (from the problem)
- What's the chance a movie gets a "good" rating if it's reviewed by a specific type of rater? These are conditional probabilities:
- P(G | C) = Probability of "good" rating GIVEN rater is "careful" = ?
- P(G | L) = Probability of "good" rating GIVEN rater is "lazy" = ?
- Use the Law of Total Probability: The overall probability of a "good" rating, P(G), is the sum of the probabilities of getting a "good" rating through each type of rater:
P(G) = (Probability of "good" if careful × Probability of careful rater) + (Probability of "good" if lazy × Probability of lazy rater)P(G) = P(G | C) * P(C) + P(G | L) * P(L) - Plug in the numbers and calculate.
Explanation: Netflix Movie Raters - Finding Overall "Good" Rating Probability
Imagine Netflix has two types of movie raters: "Careful" ones and "Lazy" ones. We want to figure out, if we pick a random movie rating, what's the overall chance that rating is "good"?
To do this, we need to consider how each type of rater contributes to the "good" ratings:
- The Careful Raters:
- How many are there? 80% of all raters are careful. (So, P(Careful) = 0.80)
- How do they rate? When a careful rater watches a movie, they rate it "good" 60% of the time. (So, P(Good | if Careful) = 0.60)
- Their contribution to "good" ratings: The portion of all movies that get a "good" rating because a careful rater reviewed them is 0.80 (the chance of getting a careful rater) × 0.60 (the chance they rate it good) = 0.48.
So, 48% of all movie ratings are "good" ratings coming from careful raters.
- The Lazy Raters:
- How many are there? The other 20% of raters are lazy. (So, P(Lazy) = 0.20)
- How do they rate? When a lazy rater watches a movie, they rate it "good" 100% of the time (they just click "good" for everything!). (So, P(Good | if Lazy) = 1.00)
- Their contribution to "good" ratings: The portion of all movies that get a "good" rating because a lazy rater reviewed them is 0.20 (the chance of getting a lazy rater) × 1.00 (the chance they rate it good) = 0.20.
So, 20% of all movie ratings are "good" ratings coming from lazy raters.
- Total Chance of a "Good" Rating: To get the overall chance that any random movie rating is "good," we just add up the contributions from both types of raters:
Contribution from Careful Raters + Contribution from Lazy Raters
= 0.48 + 0.20 = 0.68.
So, overall, there's a 68% chance that a movie is rated "good".
We want to find the overall probability that a movie is rated "good". We can use the Law of Total Probability. Let's define our events clearly:
- G: The event that a movie is rated "good". (This is what we want to find, P(G)).
- C: The event that the rater who rated the movie is a "careful" rater.
- L: The event that the rater who rated the movie is a "lazy" rater.
The problem gives us the following information (probabilities):
- The proportion of careful raters: P(C) = 0.80 (80%)
- The proportion of lazy raters: P(L) = 0.20 (20%) (Note: P(C) + P(L) = 0.80 + 0.20 = 1.00, which means these two types cover all raters and are mutually exclusive).
- If a rater is careful, the chance they rate a movie "good": P(G | C) = 0.60 (60%) (This is a conditional probability: the probability of G given C has occurred).
- If a rater is lazy, the chance they rate a movie "good": P(G | L) = 1.00 (100%) (This is also a conditional probability: the probability of G given L has occurred).
Applying the Law of Total Probability
The Law of Total Probability states that if we have a set of mutually exclusive and exhaustive events (like our rater types C and L), the total probability of another event (G) can be found by summing the probabilities of G occurring through each of those exclusive events.
The formula is: P(G) = P(G | C) * P(C) + P(G | L) * P(L)
Let's plug in the values we have:
P(G) = (0.60) * (0.80) + (1.00) * (0.20)
Now, calculate each part:
P(G) = 0.48 + 0.20
P(G) = 0.68
Final Result
The overall probability that a movie is rated "good" is 0.68.
P(Movie is rated "good") = 0.68
This means that, considering both types of raters and how often they appear, 68% of all movie ratings will be "good".
Breaking it Down Further:
- The term
P(G | C) * P(C) = 0.60 * 0.80 = 0.48represents the probability that a movie is rated by a careful rater AND is rated good by that careful rater. This is P(G and C). - The term
P(G | L) * P(L) = 1.00 * 0.20 = 0.20represents the probability that a movie is rated by a lazy rater AND is rated good by that lazy rater. This is P(G and L).
Since a movie rated "good" must have been rated by either a careful rater or a lazy rater (these are the only two types and they don't overlap for a single rating), the total probability of a "good" rating is the sum of these two joint probabilities: P(G) = P(G and C) + P(G and L).
Bayes' Twist: If a movie is randomly selected and found to be rated "good", what is the probability that it was rated by a lazy rater? (Hint: You'll need Bayes' Theorem for this: P(L | G) = [P(G | L) * P(L)] / P(G). You just calculated P(G)!).