What is the probability that an airline can accommodate all passengers who show up on a flight that has been overbooked with 195 seats but sold 200 tickets, given that the probability of an individual not showing up is 0.05?
The probability that the airline can accommodate all passengers who show up is 0.67%.
Understanding the Probability Calculation:
Probability Formula:
To calculate the probability of accommodating all passengers who show up on an overbooked flight, we use the binomial probability formula. This formula helps us determine the likelihood of a certain number of successes (passengers showing up) in a given number of trials (total number of passengers).
Key Variables:
n: Number of trials (passengers) = 200
p: Probability of success (a passenger showing up) = 0.95
X: Number of passengers who show up = 200
Calculating the Probability:
We want to find P(X = 200), which represents the probability that all 200 passengers show up without any missing their flight. Using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case:
k = 200
n = 200
p = 0.95
Final Calculation:
By substituting the values into the formula, we get:
P(X = 200) = (1) * (0.95)^200 * 1
P(X = 200) ≈ 0.0067 or 0.67%
Therefore, the probability that the airline can accommodate all passengers who show up on the overbooked flight is approximately 0.67%.
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