A box contains 55 balls numbered from 1 to 55. If 15 balls are drawn with replacement, what is the probability that at least two of them have the same number?

take prob that they are both different

first can be anything, the second must be one of the other 54
prob that they are both different = (1)(54/55) = 54/55

prob( they are NOT both different)
= prob(at least two are the same) = 1 - 54/55 = 1/55

To find the probability that at least two of the 15 balls drawn have the same number, we can subtract the probability that all of the balls have different numbers from 1.

Step 1: Find the probability that all 15 balls have different numbers.
The first ball can have any number from 1 to 55. The second ball can have any number except the number already chosen for the first ball, so it has 54 possible choices. Similarly, the third ball has 53 choices, the fourth ball has 52 choices, and so on. Therefore, the probability that all 15 balls have different numbers is given by:
P(no repeated numbers) = (55/55) * (54/55) * (53/55) * ... * (41/55)

Step 2: Calculate the probability of at least two balls having the same number.
P(at least two balls have the same number) = 1 - P(no repeated numbers)
P(at least two balls have the same number) = 1 - [(55/55) * (54/55) * (53/55) * ... * (41/55)]

Now we can calculate the probability using these steps.