For example, Chaitin's constant is both Martin-Löf random and uncomputable. Are there any examples of numbers that are Martin-Löf random but computable?
Computability is a very strong way of not being random. One easy way to understand randomness is via betting strategies (i.e. constructive martingales). These work by starting with capital 1 (dollar, say) and allowing the player to bet any amount of his current capital on what the next bit of the sequence will be. If the sequence even has an infinite c.e. subset, then the player would be guaranteed to win at those bits, so his winnings would approach $\infty$.
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