Probability and Conditional Probability Pattern - Direct Probability

Calculating probability when favorable and total outcomes are given

Probability is the foundation of data analysis. On the SAT, "direct probability" means both the favorable outcomes and the total outcomes are handed to you — your only job is to form the ratio and simplify.

The Core Formula

$P(\text{event}) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$

That's it. There are no extra steps — no computing the total, no combining categories. Both numbers are either stated in the stem or sitting in a two-way table.

What This Looks Like — Plain Stem

A bag contains 20 marbles, each labeled with a unique integer from 1 to 20. If one marble is drawn at random from the bag, what is the probability that the marble is labeled with the number 7?

A) $\dfrac{1}{20}$
B) $\dfrac{7}{20}$
C) $\dfrac{13}{20}$
D) $\dfrac{19}{20}$

Favorable outcomes: There is exactly 1 marble labeled 7.
Total outcomes: There are 20 marbles.
$P = \dfrac{1}{20}$
Answer: A

The classic trap: The wrong answers use the label's value (7) as a count of favorable outcomes, giving $\dfrac{7}{20}$. The label is just a name — only one marble carries that name.

What This Looks Like — Two-Way Table

Many direct probability questions give you a table. The row and column totals are already computed — you just need to read the right cells.

A restaurant tracked 180 pizza orders, summarized by crust type and size in the table provided.

Crust Type	Small	Medium	Large	Total
Thin	10	20	25	55
Hand-tossed	15	30	30	75
Deep Dish	5	10	15	30
Stuffed	5	10	5	20
Total	35	70	75	180

If one order is selected at random, what is the probability that it was for a deep dish pizza?

A) $\dfrac{1}{6}$
B) $\dfrac{1}{12}$
C) $\dfrac{3}{7}$
D) $\dfrac{1}{2}$

Favorable outcomes: The Deep Dish row total = 30.
Total outcomes: Grand total = 180.
$P = \dfrac{30}{180} = \dfrac{1}{6}$
Answer: A

Common Traps and Gotchas

• Using a cell instead of a row/column total. Choice B above ($\dfrac{1}{12}$) comes from using only large deep dish orders (15) instead of all deep dish orders (30). When the question says "deep dish" with no size qualifier, use the row total.

• Flipping the fraction. The denominator of a conditional probability can sometimes sneak in as a distractor. Choice D ($\dfrac{1}{2}$) is $\dfrac{15}{30}$ — the fraction of deep dish orders that are large. That answers a different question ("of deep dish orders, what fraction are large?").

• Confusing a label's value with a count. In the marble problem, "labeled 7" does not mean 7 favorable outcomes. The number on the marble is irrelevant to counting — only the fact that one marble carries that label matters.

What to Do on Test Day

• Identify the fraction immediately: Favorable outcomes go on top, total outcomes go on the bottom. Both should be directly readable from the problem or table.
• For tables, read carefully: "What is the probability of an SUV?" means use the SUV row total over the grand total. Don't accidentally use a single cell.
• Simplify your fraction by dividing numerator and denominator by their GCD. The answer choices are always in simplified form.
• Key formula: $P(\text{event}) = \dfrac{\text{favorable}}{\text{total}}$
• Probability is always between 0 and 1. If your answer is greater than 1, you've flipped something.