Math

Probability and Conditional Probability

2 min readEasy5-question drill

Probability questions on the test are usually pure counting in disguise — and they're some of the most reliable points you can grab if you know which number goes on the bottom of the fraction.

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Probability is just a measure of how likely something is, written as a fraction between 0 and 1. The formula never changes:

Probability = (number of favorable outcomes) / (total number of outcomes)

Favorable outcomes are the ones you're asked about. Total outcomes is everything that could happen. If a bag has 3 red and 7 blue marbles, the probability of drawing red is 3/10 — 3 reds out of 10 marbles total.

That's it for basic probability. The whole game is correctly counting the top number and the bottom number. You can leave answers as fractions, decimals, or percents — 3/10 = 0.3 = 30%.

Two-way tables are the test's favorite probability setup. A two-way table sorts people (or things) by two categories at once — say, age group across the top and drink preference down the side. Every probability question becomes: which cell or row or column is the favorable count, and which total is the denominator?

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Reading a two-way table

Question phrase	Denominator to use	Why
'of all people...'	Grand total	No restriction given
'of people under 30...'	Under-30 row total	Restricted to a row
'of people who chose tea...'	Tea column total	Restricted to a column

The phrasing tells you which total goes on the bottom of the fraction.

The key skill is reading which total the question wants. This is where conditional probability comes in.

Conditional probability means "the probability of A given that B is already true." The word given (or a phrase like "of the people who chose tea...") is your signal. When a question restricts you to a group, your denominator is no longer the grand total — it's the total of that restricted group.

Example using a coffee/tea table:

"What fraction of all people prefer coffee?" → denominator is everyone (the grand total).
"What fraction of people under 30 prefer coffee?" → you're now stuck inside the "under 30" row, so the denominator is the under-30 total only.

Same favorable number sometimes, different denominator. Picking the wrong denominator is the #1 mistake on these problems.

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Does the question restrict you to a subgroup ('given' / 'of those who')?

Yes ↓

→ Use that subgroup's total (row or column) as the denominator

No ↓

→ Use the grand total as the denominator

Quick rule for choosing the denominator.

Quick checklist for every probability question:

Identify the favorable outcomes (the top).
Identify the right total (the bottom) — is it everyone, or a restricted group?
Simplify or convert to the form the answer choices use.

No memorization beyond the one formula. It's all careful counting and reading the denominator correctly.

Quick check

Check your understanding with a question from this topic:

A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. If one marble is selected at random, what is the probability that it is blue?

1/2

1/5

3/10

2/5

Worked examples

Example 1

A jar contains 4 red, 6 yellow, and 10 green gumballs. If one gumball is selected at random, what is the probability that it is NOT green?

Example 2

A survey of 200 people recorded whether they preferred coffee or tea, by age group:

	Coffee	Tea	Total
Under 30	60	40	100
30 or older	50	50	100
Total	110	90	200

Of the people who prefer tea, what fraction are 30 or older?

Example 3

A class of 40 students is surveyed about whether they play a sport and whether they play an instrument:

	Instrument	No Instrument	Total
Sport	12	8	20
No Sport	6	14	20
Total	18	22	40

If a student who plays an instrument is selected at random, what is the probability that the student does NOT play a sport?

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Common pitfalls

Using the grand total when restricted

When a question says "of the people who chose X" or "given that," the denominator is that subgroup's total — NOT the overall total. Always re-read for a restricting phrase before writing the bottom number.

Mixing up row total and column total

In a two-way table, 'of the people who prefer tea' uses the tea column total, while 'of the people under 30' uses the under-30 row total. Trace your finger to the right total instead of grabbing the first one you see.

Forgetting 'NOT' / 'or' wording

'NOT green' means count everything except green; 'pizza or tacos' means add both groups. Underline these words — they change the favorable count, not the total.

Leaving the answer in the wrong form

If choices are percents, convert your fraction: 110/200 = 0.55 = 55%. A correct fraction matched to a percent choice still gets marked wrong if you stop too early.

Key takeaways

Probability = favorable outcomes / total outcomes — always a number between 0 and 1.
The hard part is choosing the correct denominator, especially in two-way tables.
'Given that' or 'of the people who...' signals conditional probability: use the subgroup's total as the denominator.
Row total vs. column total vs. grand total — trace to the exact total the question restricts you to.
Convert your answer to match the form of the choices (fraction, decimal, or percent).

Watch & learn

Curated Khan Academy walkthroughs on Probability and Conditional Probability. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Probability and Conditional Probability, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Probability and Conditional Probability

You are here

→Percentages →Two-Variable Data: Models and Scatterplots →Ratios, Rates, Proportional Relationships, and Units