A lottery draws n numbers, values 1-N , from a hopper that replaces the number after it is drawn, i.e. probability of each number appearing remains constant.
At the end, the winning number is drawn from the same hopper.
What is the probability that the number selected at first matches the winning number?
What is the probability that at least one of the numbers selected matches the winning number?
To find the probability that the number selected at first matches the winning number, we need to determine the total number of possible outcomes (denominator) and the number of favorable outcomes (numerator).
The total number of possible outcomes is N, as there are N numbers in the hopper and any of them can be selected at first.
The number of favorable outcomes is 1, as there is only one winning number.
Therefore, the probability that the number selected at first matches the winning number is 1/N.
To find the probability that at least one of the numbers selected matches the winning number, we will use the concept of complement probability. We will find the probability that none of the numbers selected matches the winning number and subtract that from 1.
Let's say the lottery draws n numbers in total (excluding the winning number).
The number of ways to select n numbers out of N is given by the combination formula, which is denoted as C(N, n) or N choose n. The formula is:
C(N, n) = N! / (n!(N-n)!), where ! represents the factorial operation.
Now, let's calculate the number of ways to select n numbers that do not match the winning number. The first number selected cannot be the winning number, so there are N-1 possibilities. Similarly, for the second number, there are N-1 possibilities. This continues until the nth number, where there are also N-1 possibilities. Therefore, the total number of ways to select n numbers that do not match the winning number is (N-1)^n.
The total number of possible outcomes (total number of ways to select n numbers) is N^n.
Therefore, the probability that none of the numbers selected matches the winning number is ((N-1)^n) / N^n.
Finally, the probability that at least one of the numbers selected matches the winning number is 1 - ((N-1)^n) / N^n.