6 ideas
In a group of just 23 people, the chance that two share a birthday exceeds one half, because what matters is the number of pairs (253), not the number of people. The surprise comes from anchoring on one's own birthday matching someone, rather than any match among all pairs.
Dice and coins have no memory, so a long run of reds at roulette does not make black more likely on the next spin. The gambler's fallacy comes from mistaking the law of large numbers for a force that corrects past imbalances. In fact early deviations are not reversed; they are only swamped by later trials.
The classical definition sets probability as favorable outcomes divided by all equally likely outcomes, so the first task is choosing a sample space whose elements really are equally likely. Many errors, such as treating the totals of two dice as equally probable, come from counting outcomes that are not symmetric.
The expected value of a bet is each payoff weighted by its probability and summed. It is the long-run average gain per play.
A probability statement is a prediction about the proportion of outcomes over many repetitions, not a guarantee about any single trial. Seen this way, rare events are expected to happen somewhere, and 'improbable' coincidences in a large population are routine.
The chance of at least one success over several trials is easiest to compute as one minus the chance of no successes at all. This is the trick behind the Chevalier de Méré's puzzle. Getting at least one six in four throws of one die is slightly better than even, but getting at least one double six in 24 throws of two dice is slightly worse than even.