To find the probability that there are more passengers than seats, we can use the binomial distribution formula.
The formula for the binomial distribution is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of having exactly k successes (in this case, passengers showing up)
- n is the number of trials (350 in this case)
- p is the probability of success (passenger showing up, which is 0.95 in this case)
- C(n, k) is the combination formula: C(n, k) = n! / (k! * (n - k)!)
To find the probability that at least 341 people with reservations show up (more passengers than seats), we need to calculate the sum of probabilities for all possible cases from 341 to 350.
P(at least 341 people with reservations show up) = P(X = 341) + P(X = 342) + ... + P(X = 350)
Let's calculate this probability step by step:
First, calculate the probability for each case:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
For k = 341:
P(X = 341) = C(350, 341) * 0.95^341 * (1 - 0.95)^(350 - 341)
Repeat this calculation for k = 342 to k = 350.
Finally, calculate the sum of all these probabilities:
P(at least 341 people with reservations show up) = P(X = 341) + P(X = 342) + ... + P(X = 350)
Using this approach, you can find the probability that there are more passengers than seats for Delta airline flights.