Calculate probability and relative frequency

The sample space is every outcome that could be picked. An outcome is favorable when it matches what the question asks about.

When you pick at random from a group,

Here’s that idea in two forms you’ll see on the SAT.

In the table, the afternoon column total is the favorable count, and the grand total is the sample space:

In the area model, a point could land anywhere in the rectangle. The $20$ regions all have the same area and $7$ are shaded, so

Counting pieces works here only because the pieces are the same size. When they aren’t, compare the actual shaded area with the actual total area. If one region covers half a board and four small regions share the other half, shading the big one gives $\frac{1}{2}$, not $\frac{1}{5}$.

Probability and relative frequency

A probability tells you how likely something is. A relative frequency tells you what fraction of the recorded data falls in a category:

On many SAT questions, the arithmetic is the same. $38$ of the $70$ attendees chose afternoon, so the relative frequency is $\frac{38}{70}$. Pick one of those attendees at random, and the probability of getting an afternoon attendee is also $\frac{38}{70}$.

A probability is always between $0$ and $1$, and it can equal $0$ or $1$. You can write it as a fraction, a decimal or a percent, unless the question asks for a particular form.

When the pick comes from the whole group, the original total is the denominator. Phrases like these tell you so:

• “selected at random from all $80$ participants”
• “one item in the box is selected”
• “a point in the figure is selected”
• “what proportion of the students”

The numerator can be one cell, several cells, a row total, a column total or an area. The denominator stays the full total.

Phrases like given that, among those who or selected from the students who change the group you’re picking from. That’s conditional probability, where the denominator shrinks to that group.

The complement of an event is the event not happening. An event and its complement cover every outcome with no overlap, so their probabilities add to $1$:

You can work with counts, too. In the opening table, $32$ of the $70$ attendees chose morning, so $70-32=38$ attendees did not choose morning, and

That matches the afternoon answer from the start, because every attendee chose one session or the other.

Neither works the same way. For “neither blue nor yellow,” count the other categories, or subtract the blue and yellow counts from the total. Don’t turn neither into one of the categories it rules out.

The bag holds $7+5+18=30$ tiles, and $30-18=12$ of them aren’t black. So the probability is

If you went straight to the $12$ tiles that aren’t black, that’s the complement doing its job.

Sometimes the SAT gives you the probability and the total, then asks for the favorable count. It’s the same relationship, used backward:

Say $35\%$ of $180$ notebooks are spiral-bound. Let $s$ be the number of spiral-bound notebooks:

Multiply both sides by the total:

This is the part-to-whole idea from ratios and proportions. The probability tells you what fraction of the full group is in the target category.

Another version gives you the probability of every category but one. Say each of $50$ club members picked red, blue or green. For a member chosen at random, $P(\text{red})=0.5$ and $P(\text{blue})=0.3$. The three colors cover every member, so their probabilities add to $1$. Green is the complement, neither red nor blue, and it gets what’s left:

So $50(0.2)=10$ members picked green. The pick is still from all $50$ members, so $50$ stays the total. On a calculator, one expression does both steps:

50(1-0.5-0.3)

Whatever the question looks like, the work goes in the same order. Here it is on the showcase example:

1. Name the sample space, which was all $80$ participants.
2. Count the favorable outcomes, the $18$ teens who completed a project.
3. Write favorable over total, $\frac{18}{80}$.
4. Simplify or convert only if the question needs it, here to $\frac{9}{40}$.

The first three steps are yours. A calculator can’t tell whether the numerator should be one cell or a whole column, whether the denominator is the original total or a smaller group, or whether a complement matches the event. With short fractions like $\frac{18}{80}$, hand work is usually faster anyway, and you can see what each number means. Once the setup is right, the Bluebook calculator is handy for awkward arithmetic or a count that takes several steps, like the green members. Type the whole expression at once, so you never round a decimal partway through and retype it.