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The probability that the number of people who show up will exceed the capacity of the plane is 0.002607722
What is the required probability?
The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
The probability that x people show up follows a binomial distribution,
[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}[/tex]
Here, n represents the number of booked people and p represents the probability that a person shows up.
Then, the probability that x people show up is:
[tex]P(x)=\frac{78!}{x!(78-x)!}*0.90^{x}*(1-0.90)^{78-x}[/tex]
So, the probability that the number of people who show up will exceed the capacity of the plane is:
P(x>76) = P(77) + P(78)
Where P(77) and P(78) are calculated as follows:
[tex]P(77)=\frac{78!}{77!(78-77)!}* 0.90^{77}* (1-0.90)^{78-77} \\ =78*0.00029969*0.10\\ =0.0023375[/tex]
[tex]P(78)=\frac{78!}{78!(78-78)!}* 0.90^{78}* (1-0.90)^{78-78} \\ =1*0.00026972*1\\ =0.00026972[/tex]
Finally, P(x>76) is:
[tex]P(x > 76) = 0.0023375 + 0.00026972\\=0.002607722[/tex]
Learn more about probability here:
https://brainly.com/question/11234923
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