When confronted with an in-flight medical emergency, pilots and crew can consult staff physicians at MedAire, an emergency facility in Tempe, Arizona. If MedAire is called, there is a 4.8 percent chance that the flight will be diverted for an immediate landing.
a.
Expected number of diversions
(if MedAire is called 8,465 times)?
b.
Probability of at least 400 diversions?
c.
Probability of fewer than 450 diversions?
To answer these questions, we need to use the concept of probability. Probability is a measure of the likelihood of an event occurring. In this case, we are interested in the probability of the flight being diverted for an immediate landing when MedAire is called.
a. Expected number of diversions (if MedAire is called 8,465 times):
The expected value, also known as the mean, can be calculated by multiplying the probability of an event by the number of times the event occurs and summing up the results.
In this case, the probability of the flight being diverted is given as 4.8%. So, if MedAire is called 8,465 times, we can calculate the expected number of diversions as follows:
Expected Number of Diversions = Probability of Diversion x Number of Times MedAire is called
Expected Number of Diversions = 0.048 x 8,465
Expected Number of Diversions ≈ 406.32 (rounded to the nearest whole number)
Therefore, the expected number of diversions is approximately 406.
b. Probability of at least 400 diversions:
To calculate the probability of at least 400 diversions, we need to sum up the probabilities of getting 400, 401, 402, and so on, up to the total number of diversions that can occur.
Since the exact distribution is not provided, we will use an approximation method called the Poisson distribution, assuming that the number of diversions follows this distribution.
Using the Poisson distribution, we can calculate the probability of at least 400 diversions as follows:
Probability of at least 400 diversions = 1 - ∑[e^(-λ) * (λ^x) / x!], where λ is the expected number of diversions (406)
Calculating this sum can be a bit complex, and if you would like a more precise answer, it may be best to use statistical software or a calculator with the Poisson distribution function. However, as an approximate answer, the probability of at least 400 diversions would be relatively high based on the given expected number (406).
c. Probability of fewer than 450 diversions:
Similar to the previous question, we can use the Poisson distribution to approximate the probability of fewer than 450 diversions.
Probability of fewer than 450 diversions = ∑[e^(-λ) * (λ^x) / x!], where λ is the expected number of diversions (406)
Again, calculating this sum exactly may be challenging without statistical software or a calculator with the Poisson distribution function. However, the probability of fewer than 450 diversions would be very close to 1 (or 100%) based on the given expected number (406).