An example: coin-tossing
(This does not mean that all possible choices of numbers within a given lottery are equally good. While the odds of winning may be the same regardless of which numbers are chosen, the expected payout is not, because of the possibility of having to share that jackpot with other players. A rational gambler might attempt to predict other players' choices and then deliberately avoid these numbers.)

What is the probability of flipping 21 heads in a row, with a fair coin? (Answer: 1 in 2,097,152 = approximately 0.000000477.) What is the probability of doing it, given that you have already flipped 20 heads in a row? (Answer: 0.5.) See Bayes' theorem.
Will you eventually come out ahead at roulette by betting double what you lost the previous time, and adding an extra amount? (Answer: given infinite time and funds, yes, you will eventually win on that color in a fair game. However, given finite time and even more finite funds, the chance exists that you will exhaust your money before winning. Regardless of the odds of a color losing (or winning) several times in a row, the probability of the ball landing on that color in a given spin is the number of that color that exist, divided by all possibilities. In the case of a Vegas roulette wheel, the chances of hitting red are 18/38, or ~.47, regardless of previous results.)
Are you more likely to win the lottery jackpot by choosing the same numbers every time or by choosing different numbers every time? (Answer: Either strategy is equally likely to win.)
Are you more or less likely to win the lottery jackpot by picking the numbers which won last week, or picking numbers at random? (Answer: Either strategy is equally likely to win.) Non-examples

Availability error
Clustering illusion
Illusion of control
Inverse gambler's fallacy
Gambler's conceit
Law of averages
Gambler's ruin
Statistical regularity