Respuesta :
Answer:
0.1150 = 11.50% probability that the sample does not contain any fibers of polymer B
Step-by-step explanation:
Hypergeometric distribution:
The probability of x sucesses is given by the following formula:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
[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]
In this question:
Fibers are chosen without replacement, which means that we use the hypergeometric distribution.
92 fibers means that [tex]N = 92[/tex]
Sample of 10 means that [tex]n = 10[/tex]
No polymeter B which means that [tex]x = 0[/tex]
17 fibers of polymeter B which means that [tex]k = 17[/tex]
a. What is the probability that the sample does not contain any fibers of polymer B
This is P(X = 0).
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 0) = h(0,92,10,17) = \frac{C_{17,0}*C_{75,10}}{C_{92,10}} = 0.1150[/tex]
0.1150 = 11.50% probability that the sample does not contain any fibers of polymer B
The probability that the sample does not contain any fibers of polymer B is 0.123
The given parameters are:
n = 92
A = 43
B = 17
C = 32
The probability of selecting a sample that contains polymer B is:
[tex]p = \frac{17}{92}[/tex]
[tex]p = 0.185[/tex]
Using binomial probability, we have:
[tex]P(x) = ^nC_x * p^x *(1 - p)^{n -x}[/tex]
The probability that a selection of 10 fibers do not contain polymer B is then calculated as:
[tex]P(0) = ^{10}C_0 * 0.185^0 *(1 - 0.185)^{10 -0}[/tex]
This gives
[tex]P(0) = 1 * 1 *(0.815)^{10}[/tex]
Evaluate the product
[tex]P(0) = 0.123[/tex]
Hence, the probability that the sample does not contain any fibers of polymer B is 0.123
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