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 can use the probability of a single event. Since there are N possible numbers that can be selected and only one winning number, the probability of selecting the winning number at first is 1/N.
To find the probability that at least one of the numbers selected matches the winning number, we can use the principle of complementary events. The complementary event to "at least one of the numbers selected matches the winning number" is "none of the numbers selected matches the winning number."
The probability that none of the numbers selected matches the winning number can be calculated as the product of the probabilities of not selecting the winning number in each draw, given that the number is replaced after each draw. In the first draw, the probability of not selecting the winning number is (N-1)/N. Similarly, in the second draw, the probability of not selecting the winning number is (N-1)/N, and so on. Since the numbers are replaced after each draw, the probabilities of not selecting the winning number remain the same for each subsequent draw.
Therefore, the probability of none of the numbers selected matching the winning number in n draws is ((N-1)/N)^n.
Now, to find the probability that at least one of the numbers selected matches the winning number, we can subtract the probability of none of the numbers selected matching the winning number from 1. Therefore, the probability of at least one of the numbers selected matching the winning number is 1 - ((N-1)/N)^n.