# 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?

Question
Upper Level 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?

2020-12-01
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:
$$\displaystyle\lambda=\frac{{19.9}}{{60}}\times{15}$$
=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:
$$\displaystyle{P}{\left({X}\ge{3}\right)}={1}-{P}{\left({X}{<}{3}\right)}$$</span>
$$\displaystyle={1}-{\left[{P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}\right]}$$
$$\displaystyle={1}-{\left[\frac{{{e}^{{-{4.7}}}{4.7}^{{0}}}}{{{0}!}}+\frac{{{e}^{{-{4.7}}}{4.7}^{{1}}}}{{{1}!}}+\frac{{{e}^{{-{4.7}}}{4.7}^{{2}}}}{{{2}!}}\right]}$$
$$\displaystyle={1}-{e}^{{-{4.7}}}{\left[{1}+{4.7}+{11.045}\right]}$$
=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.

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