in Advanced Math New Covid-19 patients arrive at the emergency room in Sultan Qaboos Hospital at a mean arrival rate of 18.8 patients per hour. Use Poisson distribution to compute the probability that at least three new patients will arrive at the emergency room within a 15-minute interval?

New Covid-19 patients arrive at the emergency room in Sultan Qaboos Hospital at a mean arrival rate of 18.8 patients per hour. Use Poisson distribution to compute the probability that at least three new patients will arrive at the emergency room within a 15-minute interval?
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Step 1
Given that,
New Covid-19 patients arrive at the emergency room in Sultan Qaboos Hospital at a mean arrival rate of 18.8 patients per hour.
Mean arrival rate in 15 minute can be calculated as follows:
$\lambda =\frac{19.9}{60}×15$
=4.7
Step 2
The probability that at least three new patients will arrive at the emergency room within a 15-minute interval can be calculated as follows:
$P\left(X\ge 3\right)=1-P\left(X<3\right)$
$=1-\left[P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right)\right]$
$=1-\left[\frac{{e}^{-4.7}{4.7}^{0}}{0!}+\frac{{e}^{-4.7}{4.7}^{1}}{1!}+\frac{{e}^{-4.7}{4.7}^{2}}{2!}\right]$
$=1-{e}^{-4.7}\left[1+4.7+11.045\right]$
=1-0.1523
=0.8477
Thus, the probability that at least three new patients will arrive at the emergency room within a 15-minute interval is 0.8477.