All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Let X = number of calls in a period of 4minutes.

X \sim \text{Po}(4 \times 3) = \text{Po}(12)

\text{P}(X=8) = 0.0655

(ii)
Let t = length of time

Let Y = number of calls in a period of t minutes.

Y \sim \text{Po}(3t)

\text{P}(Y=0) = 0.2

e^{-3t} = 0.2

t = 0.53648 minutes
t = 32 seconds

(iii)
Let W = number of calls in a working day of 12 hours.

W \sim \text{Po}(2160)

Since \lambda =2160 > 10, W \sim \text{N}(2160, 2160) approximately

\text{P}(W>2200)

= \text{P}(W>2200.5) by continuity correction

= 0.192

(iv)
Let V = number of busy days out of 6 working days.

V \sim \text{B}(6, 0.19176)

\text{P} (V=2) = 0.235

(v)
Let U = number of busy days out of 30 working days.

U \sim \text{B}(30, 0.19176)

Since n=30 is large, np-5.7528 > 5, n(1-p) = 24.2472 >5

U \sim \text{N}(5.7528, 4.6496) approximately

\text{P}(U \textless 10)

\text{P}(U \textless 9.5) by continuity correction

= 0.959

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