Suppose that on average 10% of the people with an airline-ticket doesn't show up at the flight.For a certain flight, the plane has 100 seats and 110 tickets are sold. What is the probability that everyone who shows up will have a seat?
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To determine the probability that everyone who shows up for the flight will have a seat, we need to calculate the probability that the number of people who show up is less than or equal to the number of available seats. Let's break it down step by step:
1. Find the probability that a person shows up for the flight. Given that 10% of people with tickets don't show up, the probability that a person does show up is 1 - 0.10 = 0.90.
2. Calculate the expected number of people who show up for the flight. Since 110 tickets are sold for a 100-seat plane, there could be a maximum of 110 people who show up. However, we know that on average 10% of them don't show up, so the expected number of people who do show up is 110 * 0.90 = 99.
3. Determine the probability that everyone who shows up will have a seat. This is equivalent to finding the probability that the number of people who show up for the flight is less than or equal to the number of available seats (100).
To calculate this probability, we can use binomial distribution since we have a fixed number of trials (number of people who show up) and each trial is independent (probability of showing up is constant for each person).
Using the binomial distribution formula, the probability of having k successes out of n trials can be calculated as:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- (n choose k) represents the number of ways to choose k successes out of n trials (combination).
- p is the probability of success for each trial.
- k is the number of successes (number of people who show up).
- n is the number of trials (expected number of people who show up).
In this case, we want to find the probability of having everyone who shows up (k = 99) have a seat out of a total of 100 seats (n = 100) available.
P(X <= 100) = P(X = 99) + P(X = 100)
P(X = 99) = (100 choose 99) * (0.90^99) * (0.10)^(100-99)
P(X = 100) = (100 choose 100) * (0.90^100) * (0.10)^(100-100)
Add the two probabilities to get the final result:
P(X <= 100) = P(X = 99) + P(X = 100)
Calculating these probabilities will give you the final probability that everyone who shows up will have a seat.