The exact answer depends upon three factors:
1. The table rules
For example, if the house hits soft 17 the odds are lower than if the dealer stands on soft 17.
2. How you play
For example, if you double down on an eleven against a dealer's eight, which normally the correct action, you will lose more hands than if you do not double down in that situation.
3. What you mean by losing 10 hands in a row
Some people would mean losing 10 consecutive hands without any hands that win or draw, while other people would ignore draws so that would mean losing 10 hands without winning any.
Assuming correct play and relatively common rules, with Wizard of Odds site indicates the odds for any particular hand are
That would make the odds of losing 10 hands in a row without ignoring draws about 0.08127% or 1 in 1,230.
If you want to ignore draws the odds become
That would make the odds of losing 10 hands in a row ignoring draws about 0.197% 1 in 507.