Probability Statistics

1. There are 52 cars in a deck. A deck of cards is shuffled and the top two cards are placed face down on a table.

   1. (2 pts) What is the probability that the first card is an ace?

   2. (2 pts) What is the probability that the second card is also an ace (i.e., both are

      aces)?

   3. (6 pts) What is the probability that exactly one is an ace?

2. (6 pts) An astronaut’s oxygen supply comes from two independent sources. Source A has a probability of 0.9 of working, and Source B has probability 0.8 of working. What is the probability that at least one of the sources is working?

3. When traveling from New York, Plaxico Burress uses Laguardia Airport 50% of the time, Kennedy Airport 30% of the time, and Newark Airport 20% of the time. 99% of all guns are detected by security at Laguardia. 95% of all guns are detected by security at Kennedy. 80% of all guns are detected by security at Newark.

   1. (6 pts) What is the probability that Burress is using Laguardia if his gun has been detected?

   2. (6 pts) What is the probability that Burress is using Newark if his gun has been detected?

4. 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.

   1. (2 pts) What was the probability that the number drawn was 123 on any given day?

   2. (2 pts) What was the probability that the number drawn was 666 on any given day?

   3. (2 pts) 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. (6 pts) 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?