Statistical significance - Estimating an unknown constant chance

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I am running a scenario that results in 1 of 7 possible outcomes.
The 7 outcomes are not equally likely to occur, but each iteration of the scenario has exactly the same chance for each of the 7 outcomes as the previous iterations did, so they are constant.

The goal is to estimate what the chances of these 7 outcomes are. Each iteration of the scenario takes about 5-10 seconds to run and record, MANUALLY, there is no workaround for that constraint. So it goes without saying that I need to determine how many times I need to run this scenario in order to have a statistically accurate result. I'd like each outcome to be statistically accurate within a span of about half a percentage at the very least. How many iterations would I need to achieve this, and how is it calculated?

I did 300 iterations for starters, with the following results:

+==================+============+=============+
|     Outcome      | Occurences | % occurence |
+==================+============+=============+
| 0                |         28 | 9,33%       |
+------------------+------------+-------------+
| 1                |         64 | 21,33%      |
+------------------+------------+-------------+
| 2                |         43 | 14,33%      |
+------------------+------------+-------------+
| 3                |         35 | 11,67%      |
+------------------+------------+-------------+
| 4                |         73 | 24,33%      |
+------------------+------------+-------------+
| 5                |         44 | 14,67%      |
+------------------+------------+-------------+
| 6                |         13 | 4,33%       |
+------------------+------------+-------------+
| Total occurences |        300 |             |
+------------------+------------+-------------+