Question

A metal sheet has, on the average, 5 defects per 1.0 m2
. Assuming a Poisson
distribution, calculate the probability that a 1.5 m2
piece of the metal sheet will have
at least 4 defects.

Answer 1

Answer on Question #62213 – Math – Statistics and Probability

Question

A metal sheet has, on the average, 5 defects per 1.0 m². Assuming a Poisson distribution, calculate the probability that a 1.5 m² piece of the metal sheet will have at least 4 defects.

Solution

Let X denote the number of defects in a 1.5-square-foot sheet of the metal.

Since the unit area is 1.0-square-feet, then

$$\lambda = 5 \cdot 1.5 = 7.5$$

The probability that a 1.5 m² piece of the metal sheet will have at least 4 defects is

$$\begin{array}{l} P (X \geq 4) = 1 - P (X \leq 3) = 1 - \left(P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)\right) = \\ = 1 - e ^ {- 7.5} \left(\frac {7.5^{0}}{0 !} + \frac {7.5^{1}}{1 !} + \frac {7.5^{2}}{2 !} + \frac {7.5^{3}}{3 !}\right) = 0.9409. \end{array}$$

Answer: 0.9409.

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