Trial of the Pyx
Since the early 13th century, coins struck by the Royal
Mint in England have been evaluated for their metal content on a sample basis,
in a ceremony called the Trial of the Pyx. This ceremony does not have much
meaning anymore, but there was real money on the line back in the 1700s,
because English coins were made of gold in those days. In 1799, for instance,
the procedure went like this. One hundred gold coins called guineas were chosen at
random from all of the coins made at the Mint that year, put in the Pyx (a
ceremonial box), and weighed. The Master of the Mint, who was responsible for
the quality of the coins, was allowed a margin of error, called the "remedy,"
which was set according to the manufacturing tolerances of the time.
In 1799 a guinea was supposed to weigh 128 grains (there are 360 grains in an
ounce), so the 100 guineas in the Pyx should have weighed about 12800 grains.
The remedy in those days was 1/400 of the expected amount, or 32 grains. If
the actual weight of the coins in the Pyx differed from its expected value by
more than the remedy on either the high or low side, the Master of the Mint
was exposed to serious penalties. The British government had a vested interest
in the coins not weighing too much, but the Master of the Mint had an
incentive to make them weigh less than the standard, because he was allowed to keep
the shortfall himself (as long as he was not caught by the Trial of the Pyx).
Trial of the Pyx
- If the Master of the Mint is honest and manufactures guineas that weigh exactly
128 grains on average, with a SD of 1 grain, what is the chance that he will
survive the Trial of the Pyx? To answer this
question, first build a probability model, being explicit about the population
and the sample.
- If instead he sets things up so that the guineas weigh only 127.7 grains
on average (with the same SD of 1 grain), what is the chance now that he will
survive the Trial? If he does survive, how much gold can he expect to pocket
in an average year in which he produces 100,000 guineas? Give or take how
much? Show all your work (but you don't need
to explicitly rebuild the model).