Hi there I just want to check my answer is wright, could you please help me ?
Customers are asked to evaluate hotels at an international hotel booking website. In the past, 95% of
5-star hotels received good reviews,
60% of 4-star hotels received good reviews, and 10% of 3-star hotels received
good reviews. In addition, 40% of hotels at the website are 5-star hotels, 35%
of hotels at the website are 4-star hotels, and 25% are 3-star hotels.
1. What is the probability that a hotel on the website has a good review?
2. If a hotel attains a good review, what is the probability that it will be a
5-star hotel?
3. If a hotel does not attain a good review, what is the probability that it will
be a 3-star hotel?
My answers are;
Probability of a hotel on the website has a good review.
All the good reviews; 380 + 210 + 25 = 615
Pr (G) = 615/1000 = %61.5
2) Pr (Good | 5-Star) = [Pr ( Good ∩ 5-Star) ] / [ Pr (5-Star)]
[38/100] / [95/100] = 38/95 %40
3) Number of hotels with Not-Good review.
615 out of 1000 has good 385 Not-Good
Pr ( Not-Good | 3-Star) =
[Pr ( Not-Good ∩ 3-Star)] / [ Pr( 3-Star)]
[225/1000] / [25/100] = 225/250 = %90
If a hotel attains a good review, what is the probability that it will be a 5-star hotel?
If $A$ is the event of a hotel being $5$ star and $B$ is the event of a hotel attaining good review
$P(A \cap B) = 38\% \,, \,$ as you calculated in your first answer. That is $38$ hotels out of $100$ are good as well as a $5$ star.
$P(B) = 61.5\% \, , \,$ that is $61.5$ hotels are expected to be good out of $100$.
So $P(A|B) = \frac{38}{61.5} = \frac{76}{123}$
If a hotel does not attain a good review, what is the probability that it will be a 3-star hotel?
Similar to previous one -
$100 - 61.5 = 38.5 \% \, $ hotels in total do not attain a good review. This is your $P(B)$.
$25\% - 2.5\% = 22.5 \% \, , \,$ $3$ star hotels do not attain good review. This is your $P(A \cap B)$.