Answer :
Answer:
(a) The probability that the first warped glass item is the 12th item produced is [tex]0.0009890[/tex]
(b) The probability that the first warped item is the first, second, or third item produced is [tex]0.002997001[/tex].
(c) The probability that none of the first 10 glass items produced are defective is [tex]0.9900488503187[/tex].
Step-by-step explanation:
From the question,
Glass manufacture found that 1 in every 1000 glass items produced is warped.
So, P(1 in every 1000) = [tex]\frac{1}{1000}[/tex] = [tex]0.001[/tex]
Now,
(a) Find the probability that the first warped glass item is the 12th item produced.
Here, There are occurence of data in a successive order so we use the Geometric Distribution.
Formula of Geometric Distribution = [tex](1-P)^{(x-1)}\times P[/tex]
(Where P is the probability)
Now,
P(12) = [tex](1-0.001)^{(12-1)} \times 0.001[/tex]
P(12) = 0.0009890
(b) Find the probability that the first warped item is the first, second, or third item produced.
P(1) = [tex](1-0.001)^{0} \times 0.001[/tex] = [tex]0.001[/tex]
P(2) = [tex](1-0.001)^{1} \times 0.001 = 0.000999[/tex]
P(3) = [tex](1-0.001)^{2} \times 0.001 = 0.000998001[/tex]
Use the addition rule;
[tex]P(X\leq 3) = P(1) +P(2)+P(3)[/tex]
= [tex]0.001+0.000999+0.000998001[/tex]
= [tex]0.002997001[/tex]
(c) Find the probability that none of the first 10 glass items produced are defective.
P(1) = [tex](1-0.001)^{0}\times 0.001 = 0.001[/tex]
P(2) = [tex](1-0.001)^{1}\times 0.001 = 0.000999[/tex]
using the same formula,
P(3) = [tex]0.000998001[/tex]
P(4) = [tex]0.00099700299[/tex]
P(5) = [tex]0.0009960059960[/tex]
P(6) = [tex]0.0009950099900[/tex]
P(7) = [tex]0.0009930209650[/tex]
P(8) = [tex]0.0009920279440[/tex]
P(9) = [tex]0.0009910359161[/tex]
P(10) = [tex]0.0009900448802[/tex]
Using addition rule
[tex]P(X \leq10 )= P(1)+P(2)+P(3)+P(4)+P(5)+P(6)+P(7)+P(8)+P(9)+P(10)[/tex]
= [tex]0.0099511496813[/tex]
Using the complement rule
P(X > 10) = [tex]1- P(X\leq 10)[/tex]
= [tex]1-0.0099511496813[/tex]
= [tex]0.9900488503187[/tex]