Your math is correct, but you are misinterpreting it. If you play two games of ARAM, the odds of getting Zac twice in a row are, as you say, one in 3600. However, the odds of getting ANY champion twice in a row are just 1 in 60 as the other poster stated. This is because, during the 1st game, there is a 1 in 60 chance of getting any of your 60 champions. That is, you have a 1 in 60 chance to get Zac. (You also have a 1 in 60 chance to get Ahri.) During the second game, you have a 1 in 60 chance of getting the SAME champion you had the last game. (That is, a 1 in 60 chance to get Zac again --- or, alternatively a 1 in 60 chance to get Ahri agian.)
1 in 60 times you expect to get the same champion in the second game as you got the first. 1 in 3600 times you expect to get a SPECIFIC champion in both games. Since you are only suprised at getting two champs twice in a row regardless of what champ it is, you should use the 1 in 60 figure not the 1 in 3600 figure.
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I'm not sure if you said what you meant to say, but if so, that's not how statistics work.
The odds of getting any particular champ if you own 60 would be 1/60. The odds of getting that same champ again at some point is still 1/60. But to determine the odds of getting the same champ twice in a row without any other champ in between, you multiple the denominator by itself. 60*60=3600
And if you got the same champ 3 times in a row, 606060=216,000 which has also happened to me once.