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Simple Probability
& Complements

Learn to measure the likelihood of events using fractions, and discover how complements let you find probabilities the easy way.

Pre-Algebra

What Is Probability?

Probability is a number that tells you how likely something is to happen. It always falls between \(0\) and \(1\) (or \(0\%\) and \(100\%\)).

\(P = 0\)
Impossible: the event can never happen.
\(P = 0.5\)
Equally likely: a coin flip, for example.
\(P = 1\)
Certain: the event will always happen.

Key Vocabulary

1
Experiment: An action with uncertain results (rolling a die, drawing a card).
2
Outcome: One possible result of an experiment (rolling a \(4\)).
3
Sample Space: The set of all possible outcomes (\(\{1,2,3,4,5,6\}\) for a die).
4
Event: A specific outcome or group of outcomes you care about (rolling an even number).

Probability Lives on a Number Line

0 Impossible 0.25 Unlikely 0.5 Even Chance 0.75 Likely 1 Certain Coin flip Rolling a 6

The Probability Formula

$$P(\text{event}) = \frac{\color{#4ade80}\text{number of favorable outcomes}}{\color{#fbbf24}\text{total number of outcomes}}$$

Seeing the Formula: Spinner

G G G B B A A R

Introducing the Complement

The complement of an event \(A\) is everything in the sample space that is NOT in \(A\). We write it as \(A'\) (read "A prime" or "not A").

Sample Space S A Event happens A' Event does NOT happen

The Complement Rule

This is incredibly useful: if it is hard to count the favorable outcomes directly, count the unfavorable ones and subtract from \(1\).

P(A) P(A') Together = 1 (100%)

Event vs. Complement

Event A Complement A'
Definition The outcomes you want All other outcomes
Example: "Roll > 4" \({5, 6}\) \({1, 2, 3, 4}\)
Probability \(\frac{2}{6}\) \(\frac{4}{6}\)
Sum \(\frac{2}{6} + \frac{4}{6} = 1\)

Step-by-Step Approach

1
Identify the sample space: List every equally likely outcome.
2
Count favorable outcomes: Which outcomes match the event?
3
Apply the formula: \(P = \frac{\text{favorable}}{\text{total}}\). Simplify if possible.
4
Need the complement? Subtract from \(1\): \(P(A') = 1 - P(A)\).
5
Check your answer: Is it between \(0\) and \(1\)? Does it make sense?

Example 1: Rolling a Die

A standard die has faces \(\{1, 2, 3, 4, 5, 6\}\). What is the probability of rolling an even number?

1 2 3 4 5 6 Green = favorable (even)

Example 2: Using the Complement

A bag has 10 marbles: 4 blue, 3 red, 2 green, 1 yellow. What is the probability of drawing a marble that is NOT red?

B B B B R R R G G Y

Example 3: Card Draw

A standard deck has 52 cards. What is the probability of NOT drawing a heart?

1
Sample space: \(52\) cards total.
2
Easier event first: There are \(13\) hearts, so \(P(\text{heart}) = \frac{13}{52} = \frac{1}{4}\).
3
Apply complement rule: \(P(\text{NOT heart}) = 1 - \frac{1}{4} = \frac{3}{4}\).

Example 4: "At Least One" Problems

You flip a coin 3 times. What is the probability of getting at least one head?

1
Total outcomes: \(2^3 = 8\) (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT).
2
Complement: "At least one head" is hard to count directly. Its complement is "no heads at all" = all tails = \(\{TTT\}\).
3
\(P(\text{no heads}) = \frac{1}{8}\). So \(P(\text{at least one head}) = 1 - \frac{1}{8} = \frac{7}{8}\).

Common Mistakes

✕ Probability > 1
Writing \(\frac{8}{6}\) for a die roll. Probability can never exceed 1: check your fraction.
✕ Unequal Outcomes
Using the formula when outcomes are not equally likely (e.g., a loaded die). The formula only works for fair experiments.
✕ Wrong Complement
Subtracting the number of outcomes instead of the probability: \(P(A') = 1 - P(A)\), not \(n - \text{favorable}\).

Summary Table

Concept Formula When to Use
Simple Probability \(P(A) = \frac{\text{favorable}}{\text{total}}\) Direct counting is easy
Complement \(P(A') = 1 - P(A)\) Easier to count what you DON'T want
Sum Rule \(P(A) + P(A') = 1\) Checking your work
Range \(0 \leq P \leq 1\) Validating any answer

Question 1 of 5

A jar has 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a blue marble?

Question 2 of 5

If \(P(\text{rain}) = 0.35\), what is \(P(\text{no rain})\)?

Question 3 of 5

A spinner has 12 equal sections numbered \(1\) through \(12\). What is the probability of landing on a number greater than 9?

Question 4 of 5

A bag contains 20 candies. The probability of picking a chocolate candy is \(\frac{3}{5}\). How many candies are NOT chocolate?

Question 5 of 5

You roll a standard die once. What is the probability that the result is NOT a 1?