# Probability Theory question

**Question description**

In the 1980’s, Pennsylvania held a daily lottery. People would buy lottery tickets and try to guess which 3-digit number (000 through 999) would be picked. The winning lottery number was produced from three separate machines (one for each digit) that contained 10 ping-pong balls, labelled 0 through 9. The balls were blown about in a container by a jet of air and mixed. Then one was sucked through an opening at the top of the machine.

a. What was the probability that the number drawn was 123 on any given day?

b. What was the probability that the number drawn was 666 on any given day?

c. On one day in April, 1980, the lottery number picked was 666. The
next day, the Pittsburgh newspapers reported an unusually large number of
tickets had been purchased for 666. This led to rumors that the lottery had been fixed. A major concern was 666 was a “repeat-digit” number, i.e., all of
the digits were the same. In 1000 drawings, how many repeat-digit numbers
would be expected?

d. Later, some evidence suggested a hyperdermic needle had been used to
inject paint into all of the ping-pong balls except those numbered 4 and 6 in
all of the machines. As a result, when the balls were blown around by the jet
of air, only those with 4’s and 6’s were light enough to be sucked out of the
top of the machines. What was the probability of drawing 666 given that the
lottery was fixed to produce 4’s and 6’s?

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