Statistics Flashcards: Applying The Addition Rule For Probability

What this deck covers

This deck focuses on Applying The Addition Rule For Probability, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the equivalent notation statement for $P(A\text{ and }B)$ using the intersection symbol?

Answer: $P(A\cap B)$. Intersection symbol represents outcomes in both events.

Flashcard 2: What is $P(A\text{ or }B)$ if events $A$ and $B$ are mutually exclusive?

Answer: $P(A\text{ or }B)=P(A)+P(B)$. Mutually exclusive means no overlap, so $P(A\text{ and }B)=0$.

Flashcard 3: State the Addition Rule formula for $P(A\text{ or }B)$ in terms of $P(A)$, $P(B)$, and $P(A\text{ and }B)$.

Answer: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Subtracts overlap to avoid counting shared outcomes twice.

Flashcard 4: Identify the error: A student used $P(A\text{ or }B)=P(A)+P(B)$ when $P(A\text{ and }B)\neq 0$.

Answer: Missing subtraction of $P(A\text{ and }B)$. Must subtract overlap when events aren't disjoint.

Flashcard 5: Find $P(A\text{ or }B)$ if $P(A)=\frac{1}{4}$, $P(B)=\frac{1}{3}$, and $P(A\text{ and }B)=\frac{1}{12}$.

Answer: $\frac{1}{2}$. $\frac{1}{4}+\frac{1}{3}-\frac{1}{12}=\frac{3+4-1}{12}=\frac{1}{2}$.

Flashcard 6: What value does the Addition Rule subtract to correct for double counting in $P(A\text{ or }B)$?

Answer: $P(A\text{ and }B)$. Removes the overlap where both events occur together.

Flashcard 7: Find $P(A\text{ and }B)$ if $P(A)=0.55$, $P(B)=0.30$, and $P(A\text{ or }B)=0.70$.

Answer: $0.15$. $0.55+0.30-0.70=0.15$ by rearranging.

Flashcard 8: What is the equivalent notation statement for $P(A\text{ or }B)$ using the union symbol?

Answer: $P(A\cup B)$. Union symbol represents the set of outcomes in either event.

Flashcard 9: Identify the condition that makes $P(A \text{ and } B)=0$ in the Addition Rule.

Answer: $A$ and $B$ are mutually exclusive (disjoint). Events can't happen simultaneously.

Flashcard 10: State the Addition Rule formula for $P(A \text{ or } B)$ in terms of $P(A)$, $P(B)$, and $P(A \text{ and } B)$.

Answer: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Subtract overlap to avoid counting twice.

Flashcard 11: Find $P(A\text{ or }B)$ if $A$ and $B$ are mutually exclusive with $P(A)=0.18$ and $P(B)=0.22$.

Answer: $0.40$. Mutually exclusive: just add $0.18+0.22=0.40$.

Flashcard 12: Which probability is subtracted in $P(A\text{ or }B)=P(A)+P(B)-\square$ to avoid double counting?

Answer: $P(A\text{ and }B)$. Overlap is counted in both $P(A)$ and $P(B)$.

Flashcard 13: Find $P(A\text{ or }B)$ if $A\subseteq B$, $P(A)=0.20$, and $P(B)=0.60$.

Answer: $0.60$. Since $A\subseteq B$, $A\cup B=B$.

Flashcard 14: Choose the correct statement: If $A\subseteq B$, what is $P(A\text{ or }B)$ equal to?

Answer: $P(B)$. If $A$ is subset of $B$, union equals $B$.

Flashcard 15: Find $P(A\text{ or }B)$ given $P(A)=0.25$, $P(B)=0.50$, and $P(A\text{ and }B)=0.05$.

Answer: $0.70$. Apply: $0.25+0.50-0.05=0.70$.

Flashcard 16: Find $P(A\text{ and }B)$ if $P(A)=0.48$, $P(B)=0.52$, and $P(A\text{ or }B)=0.80$.

Answer: $0.20$. $0.48+0.52-0.80=0.20$ by rearranging.

Flashcard 17: Find $P(B)$ given $P(A)=0.35$, $P(A\text{ and }B)=0.15$, and $P(A\text{ or }B)=0.65$.

Answer: $0.45$. Rearrange: $P(B)=0.65-0.35+0.15=0.45$.

Flashcard 18: Find $P(A\text{ or }B)$ given $P(A)=0.40$, $P(B)=0.30$, and $P(A\text{ and }B)=0.10$.

Answer: $0.60$. Apply: $0.40+0.30-0.10=0.60$.

Flashcard 19: Find $P(A\text{ or }B)$ if $P(A)=0.40$, $P(B)=0.35$, and $P(A\text{ and }B)=0.10$.

Answer: $0.65$. $0.40+0.35-0.10=0.65$ by Addition Rule.

Flashcard 20: Find $P(A\text{ and }B)$ if $A$ and $B$ are independent with $P(A)=0.20$ and $P(B)=0.50$.

Answer: $0.10$. Independent events: multiply $0.20(0.50)=0.10$.

Flashcard 21: What is the meaning of $P(A \text{ or } B)$ in words (inclusive or)?

Answer: Probability that at least one of $A$ or $B$ occurs. Union includes either or both events.

Flashcard 22: What is the formula for $P(A \text{ or } B)$ when $A$ and $B$ are mutually exclusive?

Answer: $P(A\text{ or }B)=P(A)+P(B)$. No overlap when events can't occur together.

Flashcard 23: Find $P(A\text{ or }B)$ given counts: $|A|=30$, $|B|=25$, $|A\cap B|=10$, total $N=100$.

Answer: $0.45$. Count unique: $(30+25-10)/100=45/100=0.45$.

Flashcard 24: Find $P(A\text{ and }B)$ given $P(A)=0.60$, $P(B)=0.50$, and $P(A\text{ or }B)=0.80$.

Answer: $0.30$. Rearrange: $P(A\text{ and }B)=0.60+0.50-0.80=0.30$.

Flashcard 25: Find $P(A)$ given $P(B)=0.55$, $P(A\text{ and }B)=0.20$, and $P(A\text{ or }B)=0.75$.

Answer: $0.40$. Rearrange: $P(A)=0.75-0.55+0.20=0.40$.

Flashcard 26: Find $P(A\text{ and }B)$ if $P(A)=\frac{3}{5}$, $P(B)=\frac{1}{2}$, and $P(A\text{ or }B)=\frac{4}{5}$.

Answer: $\frac{3}{10}$. $\frac{3}{5}+\frac{1}{2}-\frac{4}{5}=\frac{6+5-8}{10}=\frac{3}{10}$.

Flashcard 27: Interpretation check: If $P(A\text{ or }B)=0.72$, what does $0.72$ represent in context?

Answer: Chance that at least one of $A$ or $B$ occurs. Probability that event $A$ or event $B$ happens.

Flashcard 28: What condition on probabilities indicates that $A$ and $B$ are mutually exclusive?

Answer: $P(A\text{ and }B)=0$. Events cannot occur together when mutually exclusive.

Flashcard 29: Find $P(A\cup B)$ from a table where $P(A)=0.52$, $P(B)=0.27$, and $P(A\cap B)=0.12$.

Answer: $0.67$. Apply: $0.52+0.27-0.12=0.67$.

Flashcard 30: What is the complement form for $P(A\text{ or }B)$ using $P(A^c\text{ and }B^c)$?

Answer: $P(A\text{ or }B)=1-P(A^c\text{ and }B^c)$. By De Morgan's law: at least one occurs = not both fail.

Flashcard 31: What is the meaning of $P(A \text{ and } B)$ in words?

Answer: Probability that both $A$ and $B$ occur. The intersection of events $A$ and $B$.

Flashcard 32: Find $P(A\text{ or }B)$ if $P(A)=0.20$, $P(B)=0.50$, and $A$ and $B$ are mutually exclusive.

Answer: $0.70$. $0.20+0.50-0=0.70$ since mutually exclusive.

Flashcard 33: Find $P(A\text{ or }B)$ if $A$ and $B$ are independent with $P(A)=0.30$ and $P(B)=0.40$.

Answer: $0.58$. Apply: $0.30+0.40-0.30(0.40)=0.58$.

Flashcard 34: What is the formula for $P(A \text{ and } B)$ rewritten from the Addition Rule?

Answer: $P(A\text{ and }B)=P(A)+P(B)-P(A\text{ or }B)$. Rearrange the Addition Rule by solving for $P(A\text{ and }B)$.

Flashcard 35: Find $P(A\text{ or }B)$ if $P(A)=0.15$, $P(B)=0.10$, and $P(A\text{ and }B)=0$.

Answer: $0.25$. $0.15+0.10-0=0.25$ for disjoint events.

Flashcard 36: Identify the correct expression for $P(A\text{ or }B)$ when $A$ and $B$ are independent.

Answer: $P(A\text{ or }B)=P(A)+P(B)-P(A)P(B)$. For independent events, $P(A\text{ and }B)=P(A)P(B)$.

Flashcard 37: Find and correct the error: $P(A\text{ or }B)=P(A)+P(B)+P(A\text{ and }B)$.

Answer: Correct: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Should subtract, not add, the intersection term.

Flashcard 38: Find $P(A\text{ or }B)$ if $P(A)=0.90$, $P(B)=0.30$, and $P(A\text{ and }B)=0.25$.

Answer: $0.95$. $0.90+0.30-0.25=0.95$ by Addition Rule.

Flashcard 39: Find $P(A\text{ or }B)$ from counts: $n(A)=25$, $n(B)=30$, $n(A\cap B)=10$, total $n=60$.

Answer: $\frac{3}{4}$. $\frac{25+30-10}{60}=\frac{45}{60}=\frac{3}{4}$.

Flashcard 40: Which inequality must always be true for valid events: $P(A\text{ or }B)\leq \ ?$

Answer: $P(A\text{ or }B)\leq 1$. Probability cannot exceed 1 (certainty).