Using the binomial distribution, it is found that there is a 0.864 = 86.4% probability that at most 5 seniors will attend the prom.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem, we have that:
- 18% of seniors said they will attend the upcoming prom, hence p = 0.18.
- Twenty seniors chosen at random were asked if they plan to attend the prom, hence n = 20.
The probability that at most 5 seniors will attend the prom is given by:
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.18)^{0}.(0.82)^{20} = 0.0189[/tex]
[tex]P(X = 1) = C_{20,1}.(0.18)^{1}.(0.82)^{19} = 0.0829[/tex]
[tex]P(X = 2) = C_{20,2}.(0.18)^{2}.(0.82)^{18} = 0.1730[/tex]
[tex]P(X = 3) = C_{20,3}.(0.18)^{3}.(0.82)^{17} = 0.2278[/tex]
[tex]P(X = 4) = C_{20,4}.(0.18)^{4}.(0.82)^{16} = 0.2125[/tex]
[tex]P(X = 5) = C_{20,5}.(0.18)^{5}.(0.82)^{15} = 0.1493[/tex]
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = 0.0189 + 0.0829 + 0.1730 + 0.2278 + 0.2125 + 0.1493 = 0.864[/tex]
0.864 = 86.4% probability that at most 5 seniors will attend the prom.
More can be learned about the binomial distribution at