Probability of doubled digits

The other day at work I noticed roughly half the tickets assigned to me had repeated consecutive digits, such as the sevens in 47735. The probability of a five-digit number having repeated digits is easy to work out: it's the cumulative odds that any digit matches the previous digit. Given digits ABCDE,

This is close enough for an estimate, but the sample is not entirely uniform. Assigned tickets are more likely to have been created recently, and at the time the newest tickets were in the 47,700s, all of which have a doubled digit. Only one of my tickets was a 47,700, but it improved the likelihood that half my tickets would share the pattern.