An airline company is considering a new policy of booking as many as 182 persons on an airplane that can seat only 160 (Past studies have revealed that only 80% of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 182 persons. not enough seats will be available.
To estimate the probability that not enough seats will be available when the company books 182 persons, we first need to calculate the probability that more than 160 persons will show up for the flight.
Step 1: Calculate the expected number of passengers who will show up:
Expected number of passengers = Number of booked passengers * Probability of passengers showing up
Expected number of passengers = 182 * 0.8
Expected number of passengers = 145.6
Step 2: Calculate the probability that more than 160 passengers will show up:
To calculate this, we can use the binomial distribution. The binomial distribution calculates the probability of a certain number of successes (passengers showing up) in a fixed number of trials (booked passengers).
P(X > 160) = 1 - P(X ≤ 160)
P(X > 160) = 1 - Σ[P(X = x)] from x = 0 to x = 160
Here, P(X = x) is the probability of x passengers showing up for the flight.
Step 3: Calculate the probability of each possible number of passengers showing up for the flight:
To calculate P(X = x), we can use the binomial probability formula:
P(X = x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where n is the number of trials (182 booked passengers), x is the number of successes (number of passengers showing up), p is the probability of success (0.8), and (nCx) represents the number of combinations of n items taken x at a time.
Step 4: Calculate the cumulative probability:
To find P(X ≤ 160), we need to calculate the cumulative probability by summing up the probabilities of each possible number of passengers showing up from x = 0 to x = 160.
Step 5: Calculate the final probability:
Finally, subtract the cumulative probability from 1 to calculate the probability that more than 160 passengers will show up. This will give you the estimated probability that not enough seats will be available.
Please note that this estimation assumes that each passenger's decision to show up is independent of each other and that the probability of a passenger showing up is consistent for all passengers.