Question

Data from the past shows that on average, a ready-mixed concrete plant receives 100 orders for concrete every year. The maximum number of orders that the plant can fulfil each week is 2.

(a) What is the probability that in a given week the plant cannot fulfil all the placed orders?

(b) Assume the answer to part (a) is 20% (It is not; I just want to make sure that everybody uses the same number for part (b)). Suppose there are 5 of such plants. What is the probability that in a given week 2 of the plants cannot fulfill their orders?

Answer 1

In a year total number of orders = 100

Number of weeks in a year = 52 (assume a normal year)

So average # of orders/week = 100/52 = 1.923

Maximum number of orders per week = 2

So we need to find the probabilityof the No of orders/week being less than 1.92 or i.e between 0 and 1.91

The number of orders per week has a minimum value of 0,a maximum value of 2 and a mean value of 1.92

So calculate standard deviation given min =0, max =2, and mean = 1.92

Only three values are possible for # of orders 0,1,2. So we compute the std. deviation as follows, find the mean, do (mean - value)

# of orders	Mean	(Value - mean)2
0	1.92	3.6864
1	1.92	0.8464
2	1.92	0.0064
	average	1.5131
	std. dev	1.2301

So std. deviation of the # or orders = 1.2301

Now use Central Limit method to find the probability that 52 times the # of orders falls between 0 and 1, because if it is more than 1, then it has to be two

So compute the Z scores for 0 andn 1 given that Mean = 1.92 and Std. dev =1.2301

Z0 = [(0-1.92)]/1.2301/sqrt(52) = -1.92 *sqrt(52)/1.2301 = -11.2554

Z1.92 = [(1-.92)]/1.2301/sqrt(52) = 0.4689

Use normal cdf to calculate normal cdf(-11.2554, 0.4689)

You can do it in excel using normaldist function

normdist(-11.2554, using mean = 1.92, std.dev =1.2301) = 4.527 *10-27

normdist(0.4689, using mean = 1.92, std.dev =1.2301) = .11906

Therefore subtracting both we have

probability that in a given week, plant cannot full allthe placed orders = .11906

Probability of two plans having failed orders = 2/5 * .11906 = 0.0476