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 determine the probability that the number selected at first matches the winning number, you need to know the total number of possible numbers that can be drawn and the specific number that is considered the winning number.
Let's assume that the total number of possible numbers that can be drawn is N, and the winning number is a specific number X. Since the number drawn at first is replaced back into the hopper, the probability of selecting the winning number at the first draw is 1/N.
Therefore, the probability that the number selected at first matches the winning number is 1/N.
To calculate the probability that at least one of the numbers selected matches the winning number, you need to consider the complementary probability. This means calculating the probability of none of the numbers selected matching the winning number and subtracting it from 1.
The probability of not selecting the winning number at any given draw is (N-1)/N, as there would be N-1 numbers out of the N available that do not match the winning number. Since there are n draws, the probability of none of the numbers selected matching the winning number is ((N-1)/N)^n.
Therefore, the probability that at least one of the numbers selected matches the winning number is 1 - ((N-1)/N)^n.