Flip a fair coin ten times. Exactly five heads happens 24.6 per cent of the time, which is the single most likely outcome and still far from typical. Ten heads in a row happens about once in a thousand attempts.
Both facts follow from the same assumption: each flip is independent, so nothing about the previous results changes the next one.
The fallacy in both directions
After six heads, one instinct says tails is overdue and another says the coin is hot. Both are wrong for the same reason. The coin has no state. The probability of heads on the seventh flip is exactly what it was on the first.
What is true is that long runs are rare in advance. The mistake is applying that to a run already in progress. The rarity was in getting here; it says nothing about the next step.
Where the intuition comes from
Over many flips the proportion of heads does approach half, which people remember as a balancing force. It is not one. Early results are not cancelled out by later ones; they are diluted by the sheer number of flips that follow.
Six extra heads is a large imbalance in ten flips and an invisible one in ten thousand. Nothing corrects it. It just stops mattering.
Repeated attempts at an unlikely thing
Independence does not mean repetition is pointless. Something with a 25 per cent chance each attempt happens at least once in five attempts 76.3 per cent of the time. The per attempt odds never improve, but the chance of at least one success accumulates.
This is the correct version of the intuition behind chasing a streak, and it applies to the whole sequence planned in advance rather than to the next event given what has already happened.
The edge does not need a memory either
A European roulette wheel has 37 pockets and pays 35 to 1 on a single number. A fair payout would be 36 to 1. That one unit of shortfall is the entire house edge: 2.70 per cent on every simple bet. An American wheel adds a second zero, and the same payout against 38 pockets gives 5.26 per cent.
The edge applies identically to every spin regardless of what came before, which is why no betting pattern based on previous results changes it. The roulette odds calculator shows where those figures come from.
For the underlying arithmetic, the coin flip calculator handles runs and exact counts, and the probability calculator covers combined and repeated events.