Question

During the course of a Friday night, a nightclub often receives some counterfeit ten-dollar bills. At one point in the night, there are two counterfeit ten-dollar bills randomly distributed in a stack of a total of 8 ten-dollar bills (counterfeit and legitimate) in the cash register. From that point on, no additional ten-dollar bills are received, they are only paid out from the top as a change to patrons of the nightclub. What is the probability that the nightclub will have no counterfeit ten-dollar bills in its cash register if only 4 ten-dollar bills are paid out during the night?

Answer 1

If there are no counterfeit ten-dollar bills in the cash register at the end of the night, then the two counterfeit ten-dollar bills must have been paid out among the 4 ten-dollar bills during the night. Thus, the problem can be framed as follows:

Random Experiment: Drawing 4 ten-dollar bills randomly from a stack of 8 ten-dollar bills(which consists of 2 counterfeit ten-dollar bills and 6 legitimate ten-dollar bills).

Sample Space S consists of  ⁸C₄ elements.

Event E: The randomly drawn set of 4 ten-dollar bills contains the 2 counterfeit ten-dollar bills.
Then, |E| = ⁶C₂ (Out of the 8 bills, the 2 counterfeit bills can be chosen in 1 way and corresponding to this way, the other two bills can be chosen from the remaining 6 bills in ⁶C₂ ways. Hence, the operation can be done in
  1 x ⁶C₂ ways).
Thus, by the Classical Definition of Probability, the Probability of the occurence of the event E is
P(E) = |E| / |S| = ⁶C₂ / ⁸C₄ = 3/14

Hence, the probability that the nightclub will have no counterfeit ten-dollar bills in its cash register if only 4 ten-dollar bills are paid out during the night is 3/14