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How to calculate the probability of taking a seat?
Consider the simple case for just 2 seats: P(2) = 1 2 (first boarder picks his own seat with 1/2 probability) For n seats: (i) With 1 n probability, the passenger picks the seat of the first passenger, the n’th seat from the end (in which case the last passenger would definitely get his seat).
Is the probability that 1 and 2 are in the same cycle 1 / 2?
By symmetry, it’s enough to show that the probability that 1 and 2 are in the same cycle is 1 / 2. There are many ways to show this fact. For example: the probability that 1 is in a cycle of length k is 1 / n, for 1 ≤ k ≤ n.
What is the probability of customer K getting bumped?
Thus, the probability that customer k gets bumped is p(k) = 1 n∑ m ∏ ℓ = 1 1 (n + 1) − jℓ where the sum is over all sets of j values 1 < j1 < j2 < ⋯ < jm < k. That is, p(k) = 1 n∑ J ⊆ { 2, …, k − 1 } ∏ j ∈ J 1 (n + 1) − j = 1 n k − 1 ∏ j = 2 (1 + 1 (n + 1) − j) = 1 n k − 1 ∏ j = 2 (n + 2) − j (n + 1) − j = 1 n + 2 − k.
What happens when you take a random seat on a plane?
Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to board a plane that seats 100. The first person in line realizes he lost his boarding pass so when he boards he decides to take a random seat instead.
What’s the chance of getting the correct seat on a plane?
This arises from the premise that each passenger will choose the correct seat if available. Now if passengers were boarding and simply choosing seats completely at random, then the chance of you getting the correct seat would be 1 divided by the number of seats on the plane.
How can we solve the airplane probability problem?
Thankfully the underlying pattern exists whether we have a plane of 100 passengers, 1000 passengers or 10 passengers. And that, my friends, is the key to solving this bad boy → reduce the number of passengers, play out the scenarios and see what patterns we find.
How big is the probability of a plane crash?
It’s an arbitrary number. It’s too large of a number to compute the probabilities of by hand but not so large that you feel it’s impossible. Thankfully the underlying pattern exists whether we have a plane of 100 passengers, 1000 passengers or 10 passengers.