AP Statistics 2.4 Introduction to Probability Study Notes - New Syllabus
AP Statistics 2.4 Probability Rules and Complements Study Notes – New Syllabus
AP Statistics 2.4 Probability Rules and Complements Study Notes – As per latest AP Statistics Syllabus.
LEARNING OBJECTIVES
- 2.4.A Calculate probabilities for events and their complements.
ESSENTIAL KNOWLEDGE:
- 2.4.A.1 The sample space of a random process is the set of all possible nonoverlapping outcomes. The probability of the sample space is 1.
- 2.4.A.2 If all outcomes in the sample space are equally likely, the theoretical probability an event E will occur is
\( P(E)=\frac{\text{number of outcomes in event }E}{\text{total number of outcomes in the sample space}} \)
The probability of event E occurring is written as \( P(E) \). - 2.4.A.3 The probability of an event is a number between 0 and 1, inclusive.
- 2.4.A.4 The probability of the complement of an event E, which can be written as \( E’ \), \( \overline{E} \), or \( E^c \) (i.e., the probability of “not E“) is equal to \( 1-P(E) \).
2.4.A.1 Sample Space
A sample space is the complete set of all possible nonoverlapping outcomes of a random process.
Each possible outcome appears only once in the sample space, so no two outcomes overlap.
The sample space is usually represented by the symbol
\( \mathrm {S} \)
Since the sample space contains every possible outcome, the probability of the sample space is always
\( \mathrm {P(S)=1 }\)
This means that one of the outcomes in the sample space must occur whenever the random process is performed.
Key Terms
| Term | Definition |
|---|---|
| Sample Space (\(S\)) | The set of all possible nonoverlapping outcomes of a random process. |
| Outcome | One possible result of a random process. |
| Probability of the Sample Space | \(P(S)=1\) |
Examples of Sample Spaces
| Random Process | Sample Space (\(S\)) |
|---|---|
| Flip one coin | \( \{H,T\} \) |
| Roll one six-sided die | \( \{1,2,3,4,5,6\} \) |
| Draw one card from a standard deck | All 52 possible cards |
Why is \(P(S)=1\)?
Since the sample space includes every possible outcome, one of those outcomes must occur whenever the experiment is performed.
Therefore, the probability of the sample space is always
\(\mathrm { P(S)=1 }\) or $\mathrm {100\%}$
Important AP Exam Notes
- The sample space contains all possible nonoverlapping outcomes.
- The symbol for the sample space is \(S\).
- The probability of the sample space is always \(1\).
- The outcomes in a sample space should not overlap.
- Every probability question begins by identifying the correct sample space.
Common AP Exam Mistakes
| Incorrect | Correct |
|---|---|
| Leaving out possible outcomes. | Include every possible outcome. |
| Listing the same outcome twice. | Each outcome should appear only once. |
| Saying \(P(S)\) is less than 1. | Always remember \(P(S)=1\). |
Example
A fair six-sided die is rolled once.
Identify the sample space and state its probability.
▶️ Answer / Explanation
Step 1: List all possible outcomes.
\(S=\{1,2,3,4,5,6\}\)
Step 2: State the probability of the sample space.
\(P(S)=1\)
Explanation
The sample space contains every possible outcome of rolling the die.
Since one of these outcomes must occur, the probability of the sample space is always 1.
2.4.A.2 Calculating Theoretical Probability
If all outcomes in the sample space are equally likely, then the theoretical probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.
An event is a collection of one or more outcomes. The probability of an event is written as
\(\mathrm { P(E)} \)
where \(E\) represents the event.
Formula
\(\mathrm { P(E)}=\dfrac{\text{Number of Outcomes in Event }E}{\text{Total Number of Outcomes in the Sample Space}} \)
Where:
- \(P(E)\) = Probability that event \(E\) occurs.
- Number of Outcomes in Event \(E\) = Favorable outcomes.
- Total Number of Outcomes in the Sample Space = All possible equally likely outcomes.
When Can This Formula Be Used?
- All outcomes must be equally likely.
- The sample space must include every possible outcome.
- The outcomes must be nonoverlapping.
Example 1: Rolling a Fair Die
A fair six-sided die is rolled once.
Find the probability of rolling an even number.
Step 1: Write the sample space.
\(S=\{1,2,3,4,5,6\}\)
Step 2: Identify the event.
\(E=\{2,4,6\}\)
Step 3: Count the outcomes.
- Favorable outcomes = 3
- Total outcomes = 6
Step 4: Apply the formula.
\(P(E)=\dfrac{3}{6}=\dfrac12=0.50\)
Example 2: Drawing a Card
A card is selected at random from a standard deck of 52 cards.
Find the probability of drawing a King.
Step 1: Identify the event.
There are 4 Kings in the deck.
Step 2: Apply the formula.
\(P(\text{King})=\dfrac{4}{52}=\dfrac1{13}\approx0.077\)
Examples of Theoretical Probabilities
| Random Process | Event | Probability |
|---|---|---|
| Flip a fair coin | Heads | \( \frac12 \) |
| Roll a fair die | Prime number | \( \frac36=\frac12 \) |
| Draw a card | Heart | \( \frac{13}{52}=\frac14 \) |
Important AP Exam Notes
- Use the theoretical probability formula only when all outcomes are equally likely.
- The numerator counts the favorable outcomes.
- The denominator counts the total outcomes in the sample space.
- The probability of an event is written as \(P(E)\).
- Probability values must always be between 0 and 1.
Common AP Exam Mistakes
| Incorrect | Correct |
|---|---|
| Using the formula when outcomes are not equally likely. | Only use it for equally likely outcomes. |
| Using the number of favorable outcomes as the denominator. | Use the total number of outcomes in the sample space. |
| Forgetting to simplify the fraction. | Simplify the probability whenever possible. |
Example
A standard deck of 52 playing cards is shuffled thoroughly. One card is selected at random.
Calculate the probability that the selected card is either a Queen or a Heart.
▶️ Answer / Explanation
Step 1: Count the favorable outcomes.
- Queens = 4
- Hearts = 13
- Queen of Hearts is counted twice, so subtract 1.
Favorable outcomes = \(4+13-1=16\)
Step 2: Count the total number of outcomes.
52
Step 3: Apply the formula.
\(P(\text{Queen or Heart})=\dfrac{16}{52}=\dfrac4{13}\approx0.308\)
Answer
The probability of selecting either a Queen or a Heart is
\( \dfrac4{13}\approx0.308 \)
2.4.A.3 Probability Values
The probability of an event is a numerical value that measures how likely the event is to occur.
A probability is always a number between 0 and 1, inclusive.
\( 0 \le P(E) \le 1 \)
where \(P(E)\) represents the probability that event \(E\) occurs.
- \(P(E)=0\) means the event is impossible.
- \(P(E)=1\) means the event is certain.
- Probabilities between 0 and 1 represent events that are possible but not guaranteed.
Probability Scale
| Probability | Interpretation |
|---|---|
| \(0\) | Impossible event |
| Between 0 and 0.5 | Unlikely event |
| \(0.5\) | Equally likely to occur or not occur |
| Between 0.5 and 1 | Likely event |
| \(1\) | Certain event |
Examples
| Event | Probability | Interpretation |
|---|---|---|
| Rolling a 7 on a six-sided die | \(0\) | Impossible |
| Rolling an even number on a fair die | \( \frac{3}{6}=0.50 \) | Equally likely |
| Drawing any card from a deck | \(1\) | Certain |
Properties of Probability
- \(0 \le P(E) \le 1\)
- Probabilities cannot be negative.
- Probabilities cannot be greater than 1.
- A probability of 0 does not always mean an event can never occur in practice—it represents an impossible event in the probability model.
- A probability of 1 means the event will always occur.
Important AP Exam Notes
- Every probability must be between 0 and 1, inclusive.
- A probability of 0 means the event is impossible.
- A probability of 1 means the event is certain.
- Any calculated probability less than 0 or greater than 1 indicates an error in the calculation.
- Probabilities may be written as fractions, decimals, or percentages.
Common AP Exam Mistakes
| Incorrect | Correct |
|---|---|
| Probability = 1.25 | Impossible because probabilities cannot exceed 1. |
| Probability = −0.30 | Impossible because probabilities cannot be negative. |
| Confusing probability with a percentage greater than 100%. | Probability must always lie between 0 and 1 (or 0% and 100%). |
AP Exam Example
Determine whether each of the following could represent a valid probability. Justify your answer.
- \(0.82\)
- \(1.15\)
- \(-0.08\)
- \(0\)
- \(1\)
▶️ Answer / Explanation
1. \(0.82\)
Valid, because it is between 0 and 1.
2. \(1.15\)
Not valid, because probabilities cannot be greater than 1.
3. \(-0.08\)
Not valid, because probabilities cannot be negative.
4. \(0\)
Valid. It represents an impossible event.
5. \(1\)
Valid. It represents a certain event.
2.4.A.4 Complement of an Event
The complement of an event consists of all outcomes in the sample space that are not part of the event.
In other words, if event \(E\) occurs, then its complement does not occur, and if event \(E\) does not occur, then its complement does.
The complement of an event can be written as
\(E’\), \(E^c\), or \(\overline{E}\)
and is read as “not \(E\)”.
Complement Rule
The probability of the complement of an event is
\( P(E^c)=1-P(E) \)
Where:
- \(P(E)\) = Probability that event \(E\) occurs.
- \(P(E^c)\) = Probability that event \(E\) does not occur.
Why Does the Complement Rule Work?
Every outcome in the sample space must belong to either:
- Event \(E\), or
- The complement of \(E\).
Together, they include every possible outcome, so
\( P(E)+P(E^c)=1 \)
Rearranging gives the complement rule:
\( P(E^c)=1-P(E) \)
Example 1: Rolling a Die
A fair six-sided die is rolled once.
Find the probability of not rolling a 6.
Step 1: Find the probability of rolling a 6.
\( P(6)=\dfrac16 \)
Step 2: Apply the Complement Rule.
\( P(\text{not }6)=1-\dfrac16=\dfrac56 \)
Example 2: Drawing a Card
A card is selected at random from a standard deck of 52 cards.
Find the probability that the card is not a Heart.
Step 1: Find the probability of drawing a Heart.
\( P(\text{Heart})=\dfrac{13}{52}=\dfrac14 \)
Step 2: Apply the Complement Rule.
\( P(\text{not Heart})=1-\dfrac14=\dfrac34 \)
Examples of Complements
| Event \(E\) | Complement \(E^c\) |
|---|---|
| Rolling an even number | Rolling an odd number |
| Drawing a King | Drawing a card that is not a King |
| Getting Heads | Getting Tails |
Important AP Exam Notes
- The complement includes all outcomes not contained in the event.
- The complement can be written as \(E’\), \(E^c\), or \(\overline{E}\).
- Use the Complement Rule when it is easier to calculate the probability of the event than its complement (or vice versa).
- The probabilities of an event and its complement always add to 1.
Common AP Exam Mistakes
| Incorrect | Correct |
|---|---|
| Adding probabilities instead of subtracting from 1. | Use \(P(E^c)=1-P(E)\). |
| Thinking the complement is only one opposite outcome. | The complement includes every outcome not in the event. |
| Forgetting that probabilities must total 1. | Always check that \(P(E)+P(E^c)=1\). |
Example
A standard deck contains 52 cards.
Find the probability of not drawing a Queen.
▶️ Answer / Explanation
Step 1: Find the probability of drawing a Queen.
There are 4 Queens in the deck.
\( P(\text{Queen})=\dfrac{4}{52}=\dfrac1{13} \)
Step 2: Apply the Complement Rule.
\( P(\text{not Queen})=1-\dfrac1{13}=\dfrac{12}{13} \)
Answer
The probability of not drawing a Queen is
\( \dfrac{12}{13} \)
Verification
\( \dfrac1{13}+\dfrac{12}{13}=1 \)