Pity Timer and Simulation Strategy
Simulating one item at a time
It is much easier to do the math on a "per drop" basis rather than a "per lootbox" basis. Therefore, the probability of a single item being epic is 1 in 22 (0.0455) and the probability of a single item being legendary is 1 in 52 (0.0192). The probability of a single item being rare is not 1 in 4 because some loot boxes have 3 common items and 1 epic or have 3 common items and 1 legendary. Therefore, the probability of an item being rare is something less than 25%.
An analogy may help elucidate this point further. Imagine you flip two coins. The possible outcomes are HH, HT, TH, and TT. Adding up all the heads and tails would be 4 H and 4 T which is an fair coin or a 0.5 chance for heads and a 0.5 chance for tails. To simulate individual coin flips with these coins, we just have a 0.5 chance of it landing on heads and a 0.5 chance of landing on tails. However, imagine we now flip two coins but if the coins both land on tails, we re-flip them. The only possible outcomes are HH, HT, and TH. Adding the heads and tails gives 4 heads and 2 tails. Based on this, it seems like we have an unfair coin that has a 0.33 chance of landing on tails, and a 0.67 chance of landing on heads. To accurately simulate this scenario, we could either flip fair coins and reject the tails-tails combination, or we could simulate the unfair coin by having it have a 0.33 chance of tails, and a 0.67 chance of heads. By design, the output of the simulation would have 33% tails and 67% heads. However, if we simulate an unfair coin and also implemented the "no two in a row on tails" rule, we would not have an outcome of 33% tails and 67% heads. In the same light, we do not need to include a pity timer or the "one rare or better" rules because the probabilities in Table 1 already take that into account.
There are also other items you can only get from achievements or attending certain events. These items do not appear in loot boxes and are not included in this analysis.
Pity Timer?
Some gamers have suggested that there is a "pity timer" such that it is impossible to open more than around 25 loot boxes without getting a legendary. Noone has done a robust, statistical analysis of this but this article has pointed out that no known videos exist where someone opens 50+ loot boxes without getting at least a few legendaries. Based on the binomial distribution you should expect to not get any legendaries in 25 boxes 14% of the time, no legendaries in 50 boxes 2.1% of the time, and no legendaries in 100 boxes 0.045% of the time.
There are pity timers in other games made by Blizzard. In the game Diablo, one of the developers said that there was a pity timer such that a player always got a legendary drop at least every two hours of gameplay. Players of the game Hearthstone, proved the existence of a pity timer through statical analysis of 15,000 pack openings (which are similar to loot boxes). It is highly likely (but nobody has proven) that Overwatch also has some kind of pity timer. A pity timer in Overwatch would mean that the probability of getting a legendary would increase as you opened loot boxes with no legendaries. This could affect the simulation.
The language that Blizzard used to disclose the probability drops in loot boxes leaves wiggle room for a pity timer. I do not know Chinese, but the wording supposedly translates to "a Legendary item will included, on average, once in every 13.5 boxes" (source). This is a different statement than "a Legendary item will included once in every 13.5 boxes". Including the phrase "on average" allows for the probability to change depending on the number of loot boxes opened without getting a legendary, for instance. Below is a graph showing the probability of getting a legendary as a function of number of packs opened without getting a legendary (source)
However, even if there is a pity timer, the values in Table 1 have already gone through whatever effects this pity timer has on opening lots of loot boxes. Whatever the pity timer in Overwatch does, it results in a distribution of items like in Table 1. If you open a large enough number of loot boxes, the effects of the pity timer will become negligible. Using these probabilities, we do not need to take into account the possible legendary "pity timer" or the rule that one item must be rare or better