If I flip a coin 256 times, what are the odds of getting at least 162 of the same result?

For a small number of flips I would say use the binomial distribution, which is easy enough to do with a computer but not so easy to do by hand.

As the number of flips increases, the binomal distribution starts to resemble the normal distribution.

mean = 0.5 * n = 128

variance = (p)(1-p) * n

= 0.25 n

standard deviation = sqrt (variance) = 0.5 sqrt n

= 8

162 - 128 = 34

34 / 8 = 4.25 standard deviations.

"the same result" could be 162 + heads or 162 tails.

2 P(x>4.25 SD) is effectively 0. (0.002%)

if it happened, it is a sufficiently rare anomaly that you should question your assumptions. The coin loaded, or the coin flipper has some skill in flipping coins, or the count is not correct.

50/50