Respuesta :
Answer:
0.994 = 99.4% probability that a 10-cubic-foot block of the wood has at most 5 knots.
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
1.6 knots in 10 cubic feet of the wood.
This means that [tex]\mu = 1.6[/tex]
Find the probability that a 10-cubic-foot block of the wood has at most 5 knots.
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)[/tex]
In which
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-1.6}*(1.6)^{0}}{(0)!} = 0.202[/tex]
[tex]P(X = 1) = \frac{e^{-1.6}*(1.6)^{1}}{(1)!} = 0.323[/tex]
[tex]P(X = 2) = \frac{e^{-1.6}*(1.6)^{2}}{(2)!} = 0.258[/tex]
[tex]P(X = 3) = \frac{e^{-1.6}*(1.6)^{3}}{(3)!} = 0.138[/tex]
[tex]P(X = 4) = \frac{e^{-1.6}*(1.6)^{4}}{(4)!} = 0.055[/tex]
[tex]P(X = 5) = \frac{e^{-1.6}*(1.6)^{5}}{(5)!} = 0.018[/tex]
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = 0.202 + 0.323 + 0.258 + 0.138 + 0.055 + 0.018 = 0.994[/tex]
0.994 = 99.4% probability that a 10-cubic-foot block of the wood has at most 5 knots.
