the probability of an event is $(1-0.00015)^{16}$ and its negation has probability $16*0.00015-0.00015^{16}$
but their sum is: $1.00000269811092104339619658198550048851813530401806161886279350244140625$
since its not exactly $1$, does it mean i have made a mistake or can it just be the calculator? When something like this has happened before, ppl usually told me that "its very close to the actual number, so its correct". How can I determine this? What is close enough to 1?
Edit: the problem is: "The probability of a bit being corrupt is 0.00015, if there are 16 bits, what is the probability of at least one of them being corrupt?"
The sum of the probability of an event and the probability of its negation must be equal to 1. So, if you have calculated the probability of the event and its negation correctly, then their sum must be exactly 1.
The calculator that you are using may have rounding errors, which could explain why the sum is not exactly 1. However, if you are confident that you have calculated the probabilities correctly, then you should try using a different calculator or mathematical software to see if you get the same result.
Here is a way to check your calculations:
The probability of the event is (1 - 0.00015)^16 = 0.99985^16. The probability of the negation of the event is 16 * 0.00015 - 0.00015^16 = 0.00248 - 0.00015^16. The sum of the two probabilities is 0.99985^16 + 0.00248 - 0.00015^16 = 1. As you can see, the sum of the two probabilities is exactly 1, so there is no mistake in your calculations. The rounding errors in the calculator are causing the sum to be slightly less than 1.