Single-event probability

You’ve probably seen probability questions where the real trap is not the fraction , it’s listing the outcomes properly. Before you calculate anything, you need the full set of possible results for the experiment you’re doing. That full collection is called the sample space.

If your experiment is simple, a list works fast. One coin toss gives you $S=\{H,T\}$. A 4-sided die gives you $S=\{1,2,3,4\}$. If the experiment happens in stages, you may need a table or a tree diagram instead.

The sneaky part is matching the sample space to the exact experiment. If you roll two dice, $(2,5)$ and $(5,2)$ count as different outcomes because the order matters, even though both totals are 7. That catches people all the time.

How do you build a sample space?

1. Read the experiment carefully.
2. Decide whether a list, table, or tree diagram shows it best.
3. Write every possible outcome once.
4. Check that you have not merged outcomes that should stay separate.

When the experiment has two choices from different groups, you can combine them systematically. Suppose a street seller offers two sandwiches, Gatsby and Gyro, and three fruits: apple, orange, banana. Then your outcomes are:

$S=\{(\text{Gatsby, apple}),(\text{Gatsby, orange}),(\text{Gatsby, banana}),(\text{Gyro, apple}),(\text{Gyro, orange}),(\text{Gyro, banana})\}$

There are $2\times 3=6$ outcomes altogether.

Worked example

A common mistake is acting as if outcomes are automatically equally likely. They are only equally likely if the question tells you that or the setup makes it clear. Write the experiment first, then decide what kind of probability model you actually have.

Once your sample space is written, you usually care about only part of it. That smaller collection is your event. So probability is really about selecting the outcomes you want from the full list.

Take one fair die. Your sample space is $S=\{1,2,3,4,5,6\}$. If event $A$ means “rolling a 3”, then $A=\{3\}$. That’s a tiny event , just one outcome.

Now for the useful opposite idea. If the question says “not”, you often want the complement.

For the die example, if $A=\{3\}$, then $A^{\prime }=\{1,2,4,5,6\}$. Between $A$ and $A^{\prime }$, you’ve covered the whole sample space, so $A\cup A^{\prime }=S$.

Here’s the slightly surprising part: in probability, “something happens” and “it does not happen” always add to the whole of 1. That means complements are often faster than direct counting.

Worked example

This comes up all the time in questions like “not red”, “not a multiple of 4”, or “did not win”. If you can describe the opposite event more quickly, do that.