: an airline offers discounted advanced purchase fares to customers who buy tickets more than 30 days before travel and charges regular fares for tickets purchased in the last 30 days before travel. The airline notices 60% of its customers take advantage of the advanced purchase fares. The no-shows rate for people who paid regular price is 30%. The no-show rate for people who paid advance purchase was 5%. What is the probability that a customer that is a no-show had an advanced fare? What percent of all ticket holders were no shows.
huhiuh
To find the probability that a customer who is a no-show had an advanced fare, we can use Bayes' theorem. Bayes' theorem allows us to calculate the probability of an event given certain additional information or conditions.
Let's define the events:
A: Customer had an advanced fare
B: Customer is a no-show
We are given the following probabilities:
P(A) = 0.60 (60% of customers take advantage of advanced purchase fares)
P(B|A) = 0.05 (5% of customers who paid in advance are no-shows)
P(B|¬A) = 0.30 (30% of customers who paid regular price are no-shows)
We want to find P(A|B), the probability that a customer who is a no-show had an advanced fare.
Using Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
To calculate P(B), the probability of being a no-show regardless of the fare type, we can use the law of total probability. The law of total probability states that the probability of an event can be found by considering all possible ways or cases that the event can occur.
In this case, we have two cases: a customer can either have an advanced fare (A) and be a no-show (B), or have a regular fare (¬A) and be a no-show (B). Therefore:
P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A)
P(¬A) represents the probability of not having an advanced fare, which is equal to 1 - P(A).
P(B) = P(B|A) * P(A) + P(B|¬A) * (1 - P(A))
= (0.05 * 0.60) + (0.30 * 0.40)
Now we can substitute the values into the Bayes' theorem equation:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.05 * 0.60) / [(0.05 * 0.60) + (0.30 * 0.40)]
= 0.03 / (0.03 + 0.12)
= 0.03 / 0.15
= 0.20
Therefore, the probability that a customer who is a no-show had an advanced fare is 20%.
To find the percent of all ticket holders who were no-shows, we need to consider both customers who had advanced fares and those who had regular fares.
Let's define the event:
C: Customer is a no-show
We want to find P(C), the probability of a customer being a no-show.
Using the law of total probability, we can calculate P(C) by considering the two possible cases, advanced fares (A) and regular fares (¬A):
P(C) = P(C|A) * P(A) + P(C|¬A) * P(¬A)
= P(B|A) * P(A) + P(B|¬A) * P(¬A)
= 0.05 * 0.60 + 0.30 * 0.40
Now we can substitute the values:
P(C) = (0.05 * 0.60) + (0.30 * 0.40)
= 0.03 + 0.12
= 0.15
Therefore, 15% of all ticket holders were no-shows.