Page 293 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes Solution:
It is given that m = 2. Let the number of arrivals per minute be denoted by the random
variable r. The required probability is given by
2 3
e .2 0.13534 8
1. P r 3 0.18045
3! 6
2 2 r
e .2 2 4
2. P r 2 r! e 1 2 2 0.13534 5 0.6767.
r 0
2 0
e .2
3. P r 1 1 P r 0 1 1 0.13534 0.86464.
0!

Example: An executive makes, on an average, 5 telephone calls per hour at a cost which
may be taken as 2 per call. Determine the probability that in any hour the telephone calls’ cost
(i) exceeds 6, (ii) remains less than 10.

Solution:
The number of telephone calls per hour is a random variable with mean = 5. The required
probability is given by
3 5 r
e .5
1. P r 3 1 P r 3 1
r!
r 0
5 25 125 236
1 e 1 5 1 0.00678 0.7349.
2 6 6
4 5 r
e .5 5 25 125 625 1569
2. P r 4 r! e 1 5 2 6 24 0.00678 0.44324.
r 0 24

Example: A company makes electric toys. The probability that an electric toy is defective
is 0.01. What is the probability that a shipment of 300 toys will contain exactly 5 defectives?
Solution:

Since n is large and p is small, Poisson distribution is applicable. The random variable is the
number of defective toys with mean m = np = 300 × 0.01 = 3. The required probability is given by

3 5
e .3 0.04979 243
P r 5 0.10082.
5! 120

Example: In a town, on an average 10 accidents occur in a span of 50 days. Assuming that
the number of accidents per day follow Poisson distribution, find the probability that there will
be three or more accidents in a day.

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