Basic Probability

Every basic probability question is counting dressed up. The sample space is the full list of equally likely outcomes — every marble, every coin result, every card. Probability of an event is just the count of outcomes that match what you want, divided by the total count in the sample space. No formula to memorize beyond that fraction. Once you can list or count the sample space correctly, the question is already solved — the rest is arithmetic.

The trap shows up with repeated events. Flipping a coin twice does not give a one-half chance of two heads, because the two flips are independent: each one has its own one-half chance, and independent probabilities multiply, not add. Two flips give one half times one half, which is one fourth. Three flips give one half cubed, one eighth. Every extra flip cuts the chance in half again, because you are narrowing down a sample space that doubles in size with each flip, while only one outcome in it stays favorable.

Drawing from a bag works the same way but adds a twist: does the first item go back in? If you replace it, the bag resets and the second draw has the same odds as the first, so the draws are independent and multiply just like coin flips. If you do not replace it, the total count shrinks by one and the count of the item you drew shrinks too, so the second probability changes. That distinction — replacement versus no replacement — is the bridge from basic probability into conditional and dependent probability questions later.