Math Problem Statement
From experience, an airline knows that only 70% of the passengers booked for a certain flight actually show up. If passengers are randomly selected, find the probability that at most 5 of them show up. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.
Solution
We can model this situation using the binomial distribution. The number of passengers showing up follows a binomial distribution because each passenger either shows up or does not (two possible outcomes), and each passenger is independent of the others.
The binomial distribution is defined as:
where:
- is the number of passengers booked.
- is the probability that a passenger shows up (in this case, 0.70).
- is the number of passengers who show up.
- is the binomial coefficient: .
We are asked to find the probability that at most 5 passengers show up, which means calculating:
Steps:
- Identify the parameters:
- The probability of a passenger showing up:
- The probability of a passenger not showing up:
- The number of trials (passengers booked): (but this value is not provided directly, so if it's missing we cannot continue).
Could you please specify the number of passengers booked? That way, I can proceed with the calculations.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability
Formulas
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Theorems
Binomial Theorem
Suitable Grade Level
Grades 11-12 or College Level
Related Recommendation
Probability of Exceeding Plane Capacity with Normal Approximation
Calculate Probability of Passengers Showing Up on Flight
Calculate Probability Using Binomial Distribution: 80% Passenger Show-Up Rate
Calculate Probability of Passengers Showing Up for Flight
Calculate Probability of Overbooking in Binomial Distribution Using Normal Approximation