Opening
The coin has no memory, the deck of cards does not owe you a heart, and probability theory has never promised that the next toss will be kind.
Teaching narrative
Specify the sample space before counting. For three independent fair tosses there are eight equally likely sequences, and a clearly defined event is a subset of those outcomes. Counting works only because the model assigns probabilities to those sequences.
Next, distinguish combinations from permutations. Four people chosen for a committee are the same committee regardless of how we arrange their names. Four digits selected as a PIN are not necessarily the same in a different order. This distinction gives us the binomial coefficient.
A conditional probability restricts the denominator to outcomes where another event is known. That is why the probability of a positive test given disease differs from the probability of disease given a positive test. The order of the conditioning events changes the question.
Independence means learning one event occurred does not change the modeled probability of the other. Mutually exclusive events cannot occur together. Unless an event has probability zero, they are not interchangeable concepts.
Before the experiment
How many three-toss sequences have at least two heads? Does a heart and a spade drawn on one card count as independent events?
Live demonstration
Write all eight three-toss outcomes, circle the ones with at least two heads, then work through a transit-use probability tree.
Exit question
Write an example where confusing two conditional probabilities could change a consequential decision.