Probability

Overview

Probability is a medium-to-hard topic tested within the Problem-Solving and Data Analysis domain on the digital SAT. Questions ask students to compute classical probabilities, apply the complement rule, use addition and multiplication rules, calculate conditional probability from two-way tables, determine whether events are independent, and find expected values. Conditional probability questions are particularly common and require careful reading of “given that” language. These are calculator-active questions, but Desmos cannot interpret conditional logic — logical reasoning is the primary tool.

Key Points

1. Classical Probability

$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of equally likely outcomes}}$

Probability always lies between 0 and 1 (or 0% and 100%).

2. Complement Rule

$P(\text{not }A)=1-P(A)$

Particularly useful for “at least one” problems:

$P(\text{at least one})=1-P(\text{none})$

3. Addition Rule

$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$

For mutually exclusive events (A and B cannot both occur):

$P(A\text{ or }B)=P(A)+P(B)$

4. Multiplication Rule (Independent Events)

Two events A and B are independent if the occurrence of A does not change the probability of B.

$P(A\text{ and }B)=P(A)\times P(B)\text{(independent events only)}$

“With replacement” → events remain independent (the composition of the set is restored). “Without replacement” → events become dependent (denominator changes after each draw).

5. Conditional Probability

$P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}$

Trigger words on the SAT: “given that,” “if,” “provided that,” “among those who,” “of the students who”

From a two-way table: The condition restricts the sample space to one row or column.

$P(A\mid B)=\frac{\text{cell count at intersection of A and B}}{\text{row or column total for B}}$

Example:

	Passed	Failed	Total
Female	45	15	60
Male	30	10	40
Total	75	25	100

• P(Female | Passed) = 45 / 75 = 0.60 (divide by column total for “Passed”)
• P(Passed | Female) = 45 / 60 = 0.75 (divide by row total for “Female”)

These are different — the order of conditioning matters.

6. Checking Independence

Events A and B are independent if:

$P(A\mid B)=P(A)\text{or equivalently}P(A\text{ and }B)=P(A)\times P(B)$

From a table: check if the conditional distribution for A is the same across all values of B.

7. Expected Value

$E=\sum [\text{outcome}\times P(\text{outcome})]$

Example: A game pays $5withprobability0.4and$0 with probability 0.6. Expected value = 5 × 0.4 + 0 × 0.6 = $2.00

Expected value represents the long-run average outcome over many trials.

Pitfalls and Common Mistakes

Mistake 1: Dividing by the grand total instead of the conditional total. “What is the probability a student passed, given that she is female?” Students write 45/100 instead of 45/60. Fix: The “given that” condition sets your new total. Restrict the denominator to the row or column specified by the condition.

Mistake 2: Treating dependent events as independent. “Without replacement” problems: after drawing one card, the deck changes, so probabilities change. Fix: Watch for the phrases “without replacement” or “not returned.” Adjust the denominator for each subsequent draw.

Mistake 3: Confusing P(A|B) with P(B|A). These are generally not equal: P(Female|Passed) ≠ P(Passed|Female). Fix: The event after the vertical bar | is the condition. Identify which quantity is given and which is asked.

Mistake 4: Incorrectly applying “at least one.” Directly computing P(at least one) for multiple trials by summing simple probabilities can overcount overlapping events. Fix: Use the complement: P(at least one) = 1 − P(none occur in any trial).

Mistake 5: Adding probabilities for independent events instead of multiplying. “What is the probability both events occur?” Students add P(A) + P(B). Fix: For independent events occurring together, multiply: P(A and B) = P(A) × P(B).

Related Entries

• Data_Distribution_Graphs
• Statistical_Measures
• Sampling_Inference
• Scatterplots_Regression
• Ratios_Proportions

Quick Reference Card

Rule	Formula
Classical probability	P(E) = favorable / total
Complement	P(not A) = 1 − P(A)
At least one	1 − P(none)
Addition rule	P(A or B) = P(A) + P(B) − P(A and B)
Mutually exclusive	P(A or B) = P(A) + P(B)
Independent events	P(A and B) = P(A) × P(B)
Conditional (table)	Cell ÷ row or column total for the condition
Expected value	Σ [outcome × P(outcome)]
Independence check	P(A|B) = P(A)