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.

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.