an airline estimates that 97%of people booked on their flights actually show up.if the airline books 70 people on a flight for which the maximum number is 68,what is the probability that the number of people who show up will exceed the capacity of theplane
0.649
To find the probability that the number of people who show up will exceed the capacity of the plane, we need to calculate the probability of having more than 68 people show up.
Given that the airline estimates that 97% of people booked actually show up, this means that the probability of a person showing up is 0.97.
To find the probability of having more than 68 people show up out of the 70 booked, we can use the binomial distribution. The formula to calculate the probability of having k successes in n trials is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- n is the total number of trials (number of people booked)
- k is the number of successes (number of people showing up)
- p is the probability of success (probability of a person showing up)
In this case, n = 70, k > 68, and p = 0.97.
We want to calculate the probability of having k > 68 people show up:
P(k > 68) = P(k = 69) + P(k = 70)
Using the binomial formula, we can calculate each individual probability:
P(k = 69) = (70 choose 69) * (0.97)^69 * (1 - 0.97)^(70 - 69)
P(k = 70) = (70 choose 70) * (0.97)^70 * (1 - 0.97)^(70 - 70)
Calculating these probabilities:
P(k = 69) = (70! / (69!(70-69)!)) * (0.97^69) * (1 - 0.97)^(70 - 69)
P(k = 70) = (70! / (70!(70-70)!)) * (0.97^70) * (1 - 0.97)^(70 - 70)
Simplifying:
P(k = 69) = 70 * 0.97^69 * 0.03^1
P(k = 70) = 0.97^70
Calculating these probabilities:
P(k = 69) = 70 * 0.97^69 * 0.03
P(k = 70) = 0.97^70
P(k > 68) = P(k = 69) + P(k = 70)
P(k > 68) = 70 * 0.97^69 * 0.03 + 0.97^70
Therefore, the probability that the number of people who show up will exceed the capacity of the plane is approximately:
P(k > 68) = 70 * 0.97^69 * 0.03 + 0.97^70
To find the probability that the number of people who show up will exceed the capacity of the plane, we can use the concept of binomial probability.
First, let's define the variables:
p = probability of a person showing up = 0.97 (97%)
q = probability of a person not showing up = 1 - p = 1 - 0.97 = 0.03 (3%)
n = number of trials = 70 (people booked on the flight)
x = number of successes (people who show up)
To find the probability, we need to calculate the cumulative probability of all cases where x is greater than 68. This can be calculated using the formula:
P(x > 68) = P(x = 69) + P(x = 70)
The probability of x successes in n trials can be calculated using the binomial formula:
P(x) = C(n, x) * p^x * q^(n-x)
Where C(n, x) represents the combination of n items taken x at a time, given by the formula:
C(n, x) = n! / (x! * (n-x)!)
Let's calculate it step by step:
1. Calculate P(x = 69):
P(x = 69) = C(70, 69) * p^69 * q^(70-69)
= (70! / (69! * (70-69)!)) * (0.97^69) * (0.03^1)
2. Calculate P(x = 70):
P(x = 70) = C(70, 70) * p^70 * q^(70-70)
= (70! / (70! * (70-70)!)) * (0.97^70) * (0.03^0)
3. Calculate P(x > 68):
P(x > 68) = P(x = 69) + P(x = 70)
Once you plug the values into the formulas and perform the calculations, you will get the probability that the number of people who show up will exceed the capacity of the plane.