Respuesta :
0.1199 % is the probability that at least four boys are chosen.
Definition of probability
- The chance that a given event will occur. (2) : the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes.
- b : a branch of mathematics concerned with the study of probabilities.
Using the hypergeometric distribution, it is found that there is a 0.1199 = 11.99% probability that at least four boys are chosen.
The students are chosen without replacement, which is the reason why the hypergeometric distribution is used.
Hypergeometric distribution-
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N - k, n-x } }{C_{N,n} }[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
In this problem:
16 + 19 = 35 students, thus N = 35
16 are boys, thus k = 16
Five are chosen, thus n = 5
The probability that at least four are boys is:
P(x ≥ 4 ) = P(X = 4 ) + P ( X = 5)
In which
P(X = x) = h(x ,N, n, k) = [tex]\frac{C_{k,x } C_{N - k , n-k} }{C_{N,n} }[/tex]
[tex]P(x = 4) = h (4,35,5,16) = \frac{C_{16,4}C_{10,1} }{C_{35,5} } = 0.1065[/tex]
[tex]P( X= 5 ) = h (5,35,5,16) = \frac{C_{16,5} C_{19,0} }{C_{35,5} } = 0.0134[/tex]
Then-
P( X ≥ 4) = P(X = 4) + P( X = 5 ) = 0.1065 + 0.0134 = 0.1199
Learn more about probability
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