Nice coin trick, Derren Brown
Well, that was an interesting coin trick that Derren Brown showed, and the reason it works is mathematical even if the rest is bullshit. Blue's odds are 1 in 8, while Red's are 7 in 8. Why? Well, think about what they're doing.
From what I can tell, they're tossing coins until the last 3 match Blue's choice (HHH) or Red's choice (THH), then starting again with a new round of coin tosses. Now think about how Red and Blue can win. The only way Blue can win a round is if the first three coins are HHH. They can't get a HHH any other way, since that means a THH happens first and so Red wins. Therefore, Blue's odds are 1 in 8, and Red's are 7 in 8.
From what I can tell, they're tossing coins until the last 3 match Blue's choice (HHH) or Red's choice (THH), then starting again with a new round of coin tosses. Now think about how Red and Blue can win. The only way Blue can win a round is if the first three coins are HHH. They can't get a HHH any other way, since that means a THH happens first and so Red wins. Therefore, Blue's odds are 1 in 8, and Red's are 7 in 8.