The throw of the friend has always a 50% chance that the result is the same The chance of the friend is undepended on the result of the first throw . See the triangel of Blaise Pascal.
I have a roulette puzzle .
The number of corners of the France table layout is 45. What is the chance of a hit of a corner?
I like your thinking!
The answer to your question is 4 to 37 or 10.7 %
As about the quiz of WOV,if the two imprisoned persons agree to predict the same and their release depends only from knowing what the other person have predicted,then they are free men.
If the point is to guess right about the coin toss and not to guess what the other person predicted,each of them would have 50 % chance to be released.
It's quite simple,it's about binomial distribution,according to this theory,within a large amount of eight-set results,the combinations of equal distribution (4 heads and 4 tails) within each set of 8 results,would be significantly higher than the ones with the uneven distribution (regardless the sequence)
If you are interested in the subject,just google "Bernoulli trial"