Respuesta :
Answer:
k = 4
Step-by-step explanation:
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.
In this problem we have that:
[tex]n = 8, p = 0.37[/tex]. So
P(X < k) = 0.6625
k only assumes discrete values, so
[tex]P(X < k) = P(X \leq k-1) = 0.6625[/tex]
We have to find the cummulative distribution until we hit 0.6625. So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8,0}.(0.37)^{0}.(0.63)^{8} = 0.0248[/tex]
[tex]P(X \leq 0) = P(X = 0) = 0.0248[/tex]
[tex]P(X = 1) = C_{8,1}.(0.37)^{1}.(0.63)^{7} = 0.1166[/tex]
[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.0248 + 0.1166 = 0.1414[/tex]
[tex]P(X = 2) = C_{8,2}.(0.37)^{2}.(0.63)^{6} = 0.2397[/tex]
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0248 + 0.1166 + 0.2397 = 0.3811[/tex]
[tex]P(X = 3) = C_{8,3}.(0.37)^{3}.(0.63)^{5} = 0.2814[/tex]
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0248 + 0.1166 + 0.2397 + 0.2814 = 0.6625[/tex]
[tex]P(X \leq 3) = P(X < 4) = 0.665[/tex]
So k = 4
