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Hawking did not elaborate on the BBC program how the information could be extracted from the black hole.
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It's not to difficult. There are particles that "randomly" appear in the universe. But when they appear there are always two of them, a positive and a negative. What happens if one appears inside of the event horizon of a black hole and the other appears outside? One gets sucked into the black hole and is destroied while the other goes off into the universe as "Hawkings Radiation".
-Al