Question

a) Based on historical data, 9% of all logs arriving at a lumber mill are of suitable quality for use in timber frame construction. Also based on historical data, the average number of logs arriving at the mil per day is 33. Calculate, to the nearest %, the probability that 4 or more logs suitable for timber frame construction arrive at the mill on any given day. b) If 10 logs arriving at the mill are randomly selected, what is the probability, to the nearest %, that 4 or more of them will be suitable for timber frame construction? c) If a project requires 8 logs that are suitable for timber frame construction, what is the probability, to the nearest %, that enough of these logs arrive at the mill over the next 4 days?

Answer 1

5.

We can consider getting a log suitable for use in timber frame construction as success. Then we can convert our given problem into a Binomial probability distribution.

Suppose, random number X denotes the number of log arrived suitable for use in timber frame construction.

where n is number of logs arrived and p is probability to get a log suitable for use in timber frame construction in each arrival of logs.

By theory of Binomial probability distribution we have

Clearly, 9% of all logs arriving are suitable for use in timber frame construction in each arrival of logs. So, p=0.09.

(a)

Here, n=33

We can calculate the required probability using the concept of complement probability as follows.

   = 0.3455549

So, probability that 4 or more logs suitable for use in timber frame construction arrive at the mil on a given day is 34.56%.

(b)

Here, n=10

We can calculate the required probability using the concept of complement probability as follows.

   = 0.008833761

So, probability that 4 or more logs are suitable for use in timber frame construction of 10 randomly selected logs arrived is 00.88%.

(c)

Here, n=33*4=132

Project requires 8 suitable logs. So, the event of arriving at least 8 suitable logs i.e. 8 or more suitable logs is our preferable case.

Here n=132>100 is very large and p=0.09 is very small. So we can approximate binomial distribution into a Poisson distribution with parameter _..

By theory of Poisson probability distribution we have

We can calculate the required probability using the concept of complement probability as follows.

= 0.9052033

So, probability that enough of required amount of logs will arrive at the mill over the next 4 days is 90.52%.