Respuesta :
Answer:
a) 32.04% probability that overbooking occurs.
b) 40.79% probability that the flight has empty seats.
Step-by-step explanation:
For each booked passenger, there are only two possible outcomes. Either they arrive for the flight, or they do not arrive. The probability of a passenger arriving is independent of other passengers. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Our variable of interest are the 8 reservations that went for the passengers with a 48% probability of arriving.
This means that [tex]n = 8, p = 0.48[/tex]
A) Find the probability that overbooking occurs.
12 seats, 8 of which are already occupied. So overbooking occurs if more than 4 of the reservated arrive.
[tex]P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{8,5}.(0.48)^{5}.(0.52)^{3} = 0.2006[/tex]
[tex]P(X = 6) = C_{8,6}.(0.48)^{6}.(0.52)^{2} = 0.0926[/tex]
[tex]P(X = 7) = C_{8,7}.(0.48)^{7}.(0.52)^{7} = 0.0244[/tex]
[tex]P(X = 8) = C_{8,5}.(0.48)^{8}.(0.52)^{0} = 0.0028[/tex]
[tex]P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.2006 + 0.0926 + 0.0244 + 0.0028 = 0.3204[/tex]
32.04% probability that overbooking occurs.
B) Find the probability that the flight has empty seats.
Less than 4 of the booked passengers arrive.
To make it easier, i will use
[tex]P(X < 4) = 1 - (P(X = 4) + P(X > 4))[/tex]
From a), P(X > 4) = 0.3204
[tex]P(X = 4) = C_{8,4}.(0.48)^{4}.(0.52)^{4} = 0.2717[/tex]
[tex]P(X < 4) = 1 - (P(X = 4) + P(X > 4)) = 1 - (0.2717 + 0.3204) = 1 - 0.5921 = 0.4079[/tex]
40.79% probability that the flight has empty seats.
