Given: The mean rate of cars per half an hour is 3
To Determine: The probability that, in any hour, exactly 4 cars will enter the car wash
Solution
The poison formular is given as
[tex]P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}[/tex]Where
[tex]\begin{gathered} e=constant=2.718 \\ \lambda=is\text{ }an\text{ }average\text{ }rate\text{ }of\text{ }the\text{ }expected\text{ }value\text{ }and\text{ }λ=variance,alsoλ>0 \end{gathered}[/tex]If we have a mean rate of 3 cars per half an hour. Therefore in an hour the mean rate would be 6
So, substituting into the formula
[tex]P(X=4)=\frac{2.718^{-6}\times6^4}{4!}=[/tex][tex]\begin{gathered} =\frac{1296}{403.178\times24} \\ =0.13393 \\ x\approx0.1339(4decimal-place) \end{gathered}[/tex]Hence, the probability is approximately 0.1339
