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Probability of
Outcomes

Master how to calculate the likelihood of events, from single outcomes to compound scenarios, using the fundamental probability formula.

SSAT Upper Level: Quantitative

What Is Probability?

Probability measures how likely an event is to occur. It is always a number between \(0\) (impossible) and \(1\) (certain), or equivalently between \(0\%\) and \(100\%\).

\(P = 0\)
Impossible: the event can never happen.
\(P = 0.5\)
Equally likely: a fair coin landing heads.
\(P = 1\)
Certain: the event must happen every time.

Key Vocabulary

1
Experiment: An action with uncertain results (rolling a die, drawing a card).
2
Outcome: A single possible result of an experiment (\(3\) on a die).
3
Sample Space (\(S\)): The set of all possible outcomes. For a die: \(\{1,2,3,4,5,6\}\).
4
Event (\(E\)): A subset of the sample space you care about (rolling an even number: \(\{2,4,6\}\)).
5
Favorable Outcomes: Outcomes in the event that satisfy the condition.

Visualizing the Sample Space

Sample Space \(S = \{1, 2, 3, 4, 5, 6\}\)
Event \(E = \{\text{even}\}\)
\(2\)
\(4\)
\(6\)
\(1\)
\(3\)
\(5\)
\(3\) favorable outcomes
 Not in E

The Probability Formula

$$P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$
Numerator
Count the outcomes that satisfy the event condition.
Denominator
Count all equally likely outcomes in the sample space.

The Complement Rule

The complement of event \(E\), written \(E'\) or \(\overline{E}\), is everything in the sample space that is not in \(E\). Together, an event and its complement cover all outcomes:

$$P(E') = 1 - P(E)$$

Simple vs. Compound Events

Simple Event Compound Event
Definition A single outcome or a single condition Two or more simple events combined
Example Rolling a \(5\) Rolling a \(5\) and then a \(3\)
Key word one step "and" / "or"
Formula hint \(\frac{\text{favorable}}{\text{total}}\) Multiply (and) or Add (or)

Independent Events

Two events are independent if the outcome of one does not affect the other. When events are independent, multiply their probabilities:

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

Step-by-Step Approach

1
Identify the experiment. What action is being performed? (dice roll, card draw, coin flip)
2
List the sample space. Write out all equally likely outcomes.
3
Identify favorable outcomes. Which outcomes satisfy the event condition?
4
Apply the formula. \(P(E) = \frac{\text{favorable}}{\text{total}}\). Simplify the fraction.
5
Check your answer. Is it between \(0\) and \(1\)? Does it make intuitive sense?

Example 1: Single Die

A fair six-sided die is rolled once. What is the probability of rolling a number greater than \(4\)?

\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
 favorable

Example 2: Marble Bag

A bag contains \(4\) red, \(3\) blue, and \(5\) green marbles. One marble is drawn at random. What is the probability it is not green?

\(4\) red
\(3\) blue
\(5\) green

Example 3: Two Coins

Two fair coins are flipped. What is the probability of getting at least one head?

Tree Diagram
\(\text{H}\)
\(\text{T}\)
\(\text{H}\)
\(\text{T}\)
\(\text{H}\)
\(\text{T}\)
\(\text{HH}\)
\(\text{HT}\)
\(\text{TH}\)
\(\text{TT}\)
✓ at least \(1\) H
 ✕ no heads

Example 4: Cards

A standard deck has \(52\) cards. What is the probability of drawing a heart or a king?

Hearts
\(13\) cards
 Kings
\(4\) cards
\(\text{K}\hearts\)
\(1\) overlap

The "Or" Addition Rule

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$
Mutually Exclusive
Events cannot happen at the same time. Overlap = \(0\), so just add: \(P(A) + P(B)\).
Not Mutually Exclusive
Events can overlap. Subtract the overlap to avoid double-counting.

Common Mistakes

✕ Forgetting to Simplify
Writing \(\frac{3}{6}\) instead of \(\frac{1}{2}\). SSAT answers are always in simplest form.
✕ Double-Counting Overlap
Adding \(P(A) + P(B)\) for "or" problems without subtracting \(P(A \text{ and } B)\).
✕ Wrong Total Outcomes
Using \(6\) as the denominator for a \(12\)-sided die, or forgetting a deck has \(52\) cards.

Probability Cheat Sheet

Concept Formula When to Use
Basic Probability \(\frac{\text{favorable}}{\text{total}}\) Single event, one step
Complement \(1 - P(E)\) "Not," "at least one"
And (Independent) \(P(A) \times P(B)\) Two independent events both happen
Or (Mutually Excl.) \(P(A) + P(B)\) Either event, no overlap
Or (Overlap) \(P(A) + P(B) - P(A \text{ and } B)\) Either event, with overlap

Question 1 of 5

A spinner is divided into 8 equal sections numbered \(1\) through \(8\). What is the probability of spinning a prime number?

Question 2 of 5

A bag contains 6 red and 4 blue marbles. One marble is drawn. What is the probability it is not red?

Question 3 of 5

Two fair coins are flipped. What is the probability of getting exactly one head?

Question 4 of 5

A fair die is rolled and a fair coin is flipped. What is the probability of rolling a \(6\) and flipping heads?

Question 5 of 5

From a standard \(52\)-card deck, one card is drawn. What is the probability it is a spade or a face card? (Face cards: Jack, Queen, King)