Answer:
a. 61.5%; b. About 61.8%; c. About 36.4%
Step-by-step explanation:
This is a kind of question that we can solve using the Bayes' Theorem. We have here all the different conditional probabilities we need to solve this problem.
According to that theorem, the probability of a selected product attains a good review is:
(1)
In words, the probability that a selected product attains a <em>good review</em> is an <em>event </em>that depends upon the sum of the conditional probabilities that the product comes from <em>high successful product</em> P(G|H) by the probability that this product is a <em>highly successful product</em> P(H), plus the same about the rest of the probabilities, that is, P(G|M)*P(M) or the probability that the product has a good review coming from a <em>moderately successful</em> product by the probability of being moderately successful, and a good review coming from a poor successful product by the probability of being poor successful or P(G|P)*P(P).
<h3>The probability that a randomly selected product attains a good review</h3>In this way, the probability that a randomly selected product attains a good review is the result of the formula (1). Where (from the question):
P(G|H) = 95% or 0.95 (probability of receiving a good review being a highly successful product)
P(G|M) = 60% or 0.60 (probability of receiving a good review being a moderately successful product)
P(G|P) = 10% or 0.10 (probability of receiving a good review being a poorly successful product)
P(H) = 40% or 0.40 (probability of being a highly successful product).
P(M) = 35% or 0.35 (probability of being a moderately successful product).
P(P) = 25% or 0.25 (probability of being a poor successful product).
Then,
That is, <em>the probability that a randomly selected product attains a good review</em> is 61.5%.
<h3>The probability that a new product attains a good review is a highly successful product</h3>We are looking here for P(H|G). We can express this probability mathematically as follows (another conditional probability):
We can notice that the probability represents a fraction from the probability P(G) already calculated. Then,
Then, the probability of a product that attains a good review is indeed a highly successful product is about 0.618 or 61.8%.
<h3>The probability that a product that <em>does not attain </em>a good review is a moderately successful product</h3>The probability that a product does not attain a good review is given by a similar formula than (1). However, this probability is the complement of P(G). Mathematically:
P(NG|H) = 1 - P(G|H) = 1 - 0.95 = 0.05
P(NG|M) = 1 - P(G|M) = 1 - 0.60 = 0.40
P(NG|P) = 1 - P(G|M) = 1 - 0.10 = 0.90
So
Which is equal to
P(NG) = 1 - P(G) = 1 - 0.615 = 0.385
Well, having all this information at hand:
Then, the <em>probability that a new product does not attain a good review and it is a moderately successful product is about </em>0.364 or 36.4%.
