Answer:
[tex]P(X = 2) = 0.2613[/tex]
Step-by-step explanation:
Given
[tex]p = 4\%[/tex] --- proportion
[tex]x = 2[/tex] --- defective motors
[tex]n = 60[/tex] --- sample size
Required
Determine the probability that exactly 2 is defective
This follows a Poisson distribution and will be solved using:
[tex]P(X = x) = \frac{e^{-u} u^x}{x!}[/tex]
Where
u = Expected number of occurrence, and it is calculated as:
[tex]u = np[/tex]
[tex]u = 60 * 4\%[/tex]
[tex]u = 60 * 0.04[/tex]
[tex]u = 2.4[/tex]
So:
[tex]P(X = x) = \frac{e^{-u} u^x}{x!}[/tex] becomes
[tex]P(X = 2) = \frac{e^{-2.4}2.4^{2}}{2!}[/tex]
[tex]P(X = 2) = \frac{e^{-2.4}* 5.76}{2}[/tex]
[tex]P(X = 2) = e^{-2.4}* 2.88[/tex]
[tex]P(X = 2) = 0.09071795328* 2.88[/tex]
[tex]P(X = 2) = 0.26126770544[/tex]
[tex]P(X = 2) = 0.2613[/tex]
Hence, the probability that exactly 2 out of 60 will be defective is 0.2613
