Question

In a certain state's lottery, 26 balls numbered 1 through 26 are placed in a machine and 6 of them are drawn at random. If the 6 numbers drawn match the numbers that a player had chosen, the player wins $30,000. In this lottery, the order the numbers are drawn in does matter. Compute the probability that you win the prize if you purchase a single lottery ticket.

a) 1/230,230
b) 1/14,950
c) 1/10,400
d) 1/2,330

Answer 1

The probability of winning the state lottery where the order matters is calculated using permutations. The total number of outcomes for drawing 6 numbers from 26 is 26P6, and the probability of winning is 1 divided by this number of outcomes.

Step-by-step explanation:

The probability of winning the lottery in the given scenario is a classic example of a combination problem in probability. Since the order in which the balls are drawn matters, we calculate the probability using permutations. With 26 balls and 6 to be chosen, the number of different combinations in which 6 numbers can be drawn from 26 is calculated using the permutation formula nPr = n! / (n - r)!, where n is the total number of options (26 in this case), and r is the number of selections made (6 in this case).

Thus, we can calculate the total number of possible outcomes as 26P6. Once we have this number, we know that there is only one successful outcome—that which matches a player's chosen numbers exactly. Therefore, the probability of winning is 1 divided by the total number of possible outcomes.