Answer :
We have to calculate the probability of having at least 7 defects in a 26 sq ft metal sheet.
The number of defects is modeled by a Poisson distribution with an average umber of defects of 4 defects per 16 sq ft.
We can write the parameter rate as:
[tex]r=\frac{4\text{ defects}}{16\text{ sq ft}}=0.25\text{ defect per sq ft}[/tex]Then, we have to calculate the probability of having at least 7 defects in a 26 sq ft metal sheet.
The probability can be calculated as:
[tex]P(k\ge7)=1-P(k<7)=1-\sum ^6_{k=0}P(k)[/tex]The parameter, average number of defects, for this metal sheet can be calculated as:
[tex]r\cdot S=\frac{0.25\text{ defects}}{\text{sqft}}\cdot26\text{ sqft}=6.5\text{ defects}[/tex]Then, we can calculate all the probabilities from k=0 to k=6 as:
[tex]\begin{gathered} P(0)=6.5^0\cdot e^{-6.5}/0!=1*0.0015/1=0.002 \\ P(1)=6.5^1\cdot e^{-6.5}/1!=6.5*0.0015/1=0.01 \\ P(2)=6.5^2\cdot e^{-6.5}/2!=42.25*0.0015/2=0.032 \\ P(3)=6.5^3\cdot e^{-6.5}/3!=274.625*0.0015/6=0.069 \\ P(4)=6.5^4\cdot e^{-6.5}/4!=1785.0625*0.0015/24=0.112 \\ P(5)=6.5^5\cdot e^{-6.5}/5!=11602.9063*0.0015/120=0.145 \\ P(6)=6.5^6\cdot e^{-6.5}/6!=75418.8906*0.0015/720=0.157 \end{gathered}[/tex]Then, we can calculate:
[tex]\begin{gathered} P(x\ge7)=1-P(x<7) \\ P(x\ge7)=1-(0.002+0.01+0.032+0.069+0.112+0.145+0.157) \\ P(x\ge7)=1-0.525 \\ P(x\ge7)=0.475 \end{gathered}[/tex]The probability of having 7 or more defects in the 26 sq ft metal sheet is 0.475.
