Question

4). Commemorative coins are being struck at the local foundry. A gold blank (a solid gold disc with no markings on it) is inserted into a hydraulic press and the obverse design is pressed onto one side of the disc (this step fails with probability 0.15). The work is examined and if the obverse pressing is good, the coin is put into a second hydraulic press and the reverse design is imprinted (this step fails with probability 0.08). The completed coin is now examined and if of sufficient quality is passed on for finishing (cleaning, buffing, and so on). Twenty gold blanks are going to undergo pressing for this commemorative coin. Assume that all pressings are independent of each other.

4a: What are the mean and variance of the number of good coins manufactured?

4b: If the blanks cost $300 each and the labor to produce the finished coins costs $3,000, what is the probability that the production cost to make the 20 coins (labor and materials) can be recovered by selling the coins for $500 each?

Answer 1

4.

A coin will be good if its both sides are pressed correctly i.e. both steps do not fail.

So, probability that a good coin is manufactured

(a)

We can model the given situation using Binomial distribution as follows.

Suppose, random variable X denotes number of good coins.

A coin to be good or not is independent of other coins.

We define getting a good coin as success.

For binomial distribution we have,

Hence, mean and variance are 15.64 and 3.40952 respectively.

(b)

Suppose, random variable Y denotes total selling price.

We can approximate the distribution of Y to normal distribution.

Production cost of making 20 coins

Probability that production cost to make these 20 coins can be recovered by selling the coins is given by

   [Using R-code '1-pnorm(1.278102)']