Question

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. What are the mean and variance of the number of good coins manufactured? 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? (The $3,000 labor figure is the fixed cost to process
the 20 gold blanks -- some will be good some not.)

Answer 1

Answer :

the obverse design is pressed onto one side of the disc (this step fails with probability 0.15). This is same as

The probability that the step of pressing the obverse design fails is 0.15

The probability that the step of pressing the obverse design succeeds is

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). This is same as

The conditional probability that the step of pressing the Reverse design fails given that the obverse design succeeds is 0.08

The conditional probability that the step of pressing the Reverse design succeeds given that the obverse design succeeds is

The probability that a good coin is manufactured using any given blank is same as the probability that obverse design is good and the reverse design is good

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

Let X be the number of good coins manufactured using twenty gold blanks. We can say that X has a Binomial distribution with parameters, number of trials (number of blanks) n=20 and success probability (The probability that a good coin is manufactured using any given blank) p=0.782

The probability that X = x good coins are manufactured is

The expected value of X (using the formula for Binomial distribution) is

The variance of X is (using the formula for Binomial distribution)

ans: the mean of the number of good coins manufactured is 15.64

and variance of the number of good coins manufactured is 3.4095

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?

There are 20 blanks, the total cost of the blanks is 20*300=$6,000

The total labor cost to produce the coins is $3,000 (irrespective of if the coins fail in the 1st step or 2nd step)

The total cost is $6000 + $3000 = $9000

Each good coin can be sold at $500. If X is the number of good coins produced, to recover $9000, we need to produce

That is, to recover the cost, we need to produce at least 18 good coins

The probability of manufacturing at least 18 good coins is

ans: the probability that the production cost to make the 20 coins (labor and materials) can be recovered by selling the coins for $500 each is 0.1561