[Harri (OP)]

I'm hopeless at mathematics and certainly cannot work out the odds of something occurring and reoccurring so I hope someone can help. It's not a life or death question I'm just curious.

I have a digital clock next to my bed and have woken up and looked at the time to find 3:33. Twice. I have also woken up to see 5.55 and  4.44. This might be a common occurrence for some but I thought it unusual and wondered what the odds were of waking and seeing 3 digits with the same number? 

[Bored chemist]

Whatever the first digit is, there's only 1 minute in each hour when the other two digits will be the same as it. (1:11  2:22  etc)
So the odds are (to a good approximation) 1 in 60 for any given glance at the clock. I'm sort of assuming that 12:12 would count too- or that you wouldn't be in bed that early :-).
So if you look at the clock a couple of times in the night, you would see the pattern of 3 digits about a dozen times a year.

It probably "feels" more common than that, because you remember the "interesting" times like 3:33 (and maybe 1:23 or 5:43) but forget the "boring" ones when it's 3:36 or whatever.

It's possible that you might also consider it "odd" to see the time as exactly on the hour- say 3:00
Or where the numbers form a sequence- like 1:23 or 6:54

So that's 4 minutes each hour- odds of 1 in 15.
Since you sleep every night, that could easily mean seeing a "peculiar" number twice a month.