This is kind of a neat riddle problem because there is a bit of confounding information.
The fact that you are looking to recreate the first 10 flips actually means those first 10 flips don’t matter at all. Yet some people are likely to fall in the trap presented. The problem actually is just “what is the probability of flipping 10 coins in a specific sequence?”
As many others have said, it works out to 1/1024 since the probability of each flip is 1/2 and then multiply them all together.
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