> For example, for an event with two possible outcomes, where each outcome is likely to happen 50% of the time, the probability of the same outcome occurring six times in a row is less than 3 in 1000
This is wrong. The probability of, e.g. getting heads 6 times in a row is 1/2^6 which is 1/64. The probability of getting 6 heads in a row or six tails in a row is 1/2^5, which is 1/32.
> For example, for an event with two possible outcomes, where each outcome is likely to happen 50% of the time, the probability of the same outcome occurring six times in a row is less than 3 in 1000
This is wrong. The probability of, e.g. getting heads 6 times in a row is 1/2^6 which is 1/64. The probability of getting 6 heads in a row or six tails in a row is 1/2^5, which is 1/32.