Statistics Flashcards: Applying The Multiplication Rule For Probability

What this deck covers

This deck focuses on Applying The Multiplication 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: In a uniform model with $\#(S)=50$, $\#(A)=20$, $\#(A \cap B)=5$, find $P(A \cap B)$.

Answer: $\frac{1}{10}$. Count favorable outcomes over total: $\frac{5}{50} = \frac{1}{10}$.

Flashcard 2: Which expression equals the probability that both events occur: $P(A \text{ and } B)$?

Answer: $P(A \cap B)$. "And" in probability means intersection.

Flashcard 3: What is $P(A \cap B)$ if $P(B)=0.20$ and $P(A\mid B)=0.50$?

Answer: $0.10$. Use $P(A \cap B) = P(B)P(A|B) = 0.20 \times 0.50$.

Flashcard 4: State the general Multiplication Rule for the probability of $A \cap B$.

Answer: $P(A \cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$. Expresses joint probability using conditional probability.

Flashcard 5: If $P(A \cap B)=0$, what must be true about $A$ and $B$?

Answer: $A$ and $B$ are mutually exclusive (cannot occur together). Zero intersection means events have no common outcomes.

Flashcard 6: What condition makes the simplified rule $P(A \cap B)=P(A)P(B)$ valid?

Answer: $A$ and $B$ are independent. Independence allows factoring joint probability into product.

Flashcard 7: In a uniform probability model, how do you compute $P(B\mid A)$ from counts?

Answer: $P(B\mid A)=\frac{\#(A \cap B)}{\#(A)}$. Restricts sample space to outcomes where $A$ occurred.

Flashcard 8: State the definition of conditional probability $P(B\mid A)$ in terms of $P(A \cap B)$ and $P(A)$.

Answer: $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$. Ratio of joint probability to marginal probability of $A$.

Flashcard 9: Find $P(A \cap B)$ given $P(A)=0.4$ and $P(B\mid A)=0.25$.

Answer: $P(A \cap B)=0.10$. Apply multiplication rule: $0.4 \times 0.25 = 0.10$.

Flashcard 10: In a uniform probability model, how do you compute $P(E)$ from counts?

Answer: $P(E)=\frac{\#(E)}{\#(S)}$. Uniform model means all outcomes are equally likely.

Flashcard 11: In a uniform model, what is $P(B\mid A)$ in terms of counts $|A\cap B|$ and $|A|$?

Answer: $P(B\mid A)=\frac{|A\cap B|}{|A|}$. Count outcomes in both $A$ and $B$ divided by outcomes in $A$.

Flashcard 12: Two fair coins are flipped. Let $A=\{\text{first is H}\}$ and $B=\{\text{exactly one H}\}$. Find $P(A \cap B)$.

Answer: $P(A \cap B)=\frac{1}{4}$. Only HT satisfies both conditions out of 4 outcomes.

Flashcard 13: Two fair coins are flipped. Let $A=\{\text{first is H}\}$ and $B=\{\text{exactly one H}\}$. Find $P(B\mid A)$.

Answer: $P(B\mid A)=\frac{1}{2}$. Given first H, only HT has exactly one H out of HH, HT.

Flashcard 14: In a uniform model, if $|A|=10$ and $|A\cap B|=4$, find $P(B\mid A)$.

Answer: $P(B\mid A)=\frac{4}{10}=0.4$. Four outcomes satisfy both conditions out of 10 in $A$.

Flashcard 15: Compute $P(B\mid A)$ given $P(A \cap B)=0.12$ and $P(A)=0.3$.

Answer: $P(B\mid A)=0.4$. Divide joint probability by marginal: $0.12 \div 0.3 = 0.4$.

Flashcard 16: Which formula gives $P(B\mid A)$ in terms of $P(A \cap B)$ and $P(A)$?

Answer: $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$. Rearranges the multiplication rule to isolate conditional probability.

Flashcard 17: In a uniform probability model with $N$ equally likely outcomes, what is $P(E)$ for an event with $|E|$ outcomes?

Answer: $P(E)=\frac{|E|}{N}$. Favorable outcomes divided by total outcomes in uniform model.

Flashcard 18: Identify the independence criterion stated using conditional probability.

Answer: $P(B\mid A)=P(B)$ (with $P(A)>0$). Independence means conditioning doesn't change probability.

Flashcard 19: What equivalent Multiplication Rule expression gives $P(A \cap B)$ starting with $P(B)$?

Answer: $P(A \cap B)=P(B)P(A\mid B)$. Alternative form using $P(B)$ first, then $P(A|B)$.

Flashcard 20: Which formula gives $P(A \cap B)$ using $P(A)$ and $P(B\mid A)$?

Answer: $P(A \cap B)=P(A)P(B\mid A)$. Direct application of the multiplication rule.

Flashcard 21: Find $P(A \cap B)$ given $P(B)=0.3$ and $P(A\mid B)=0.5$.

Answer: $P(A \cap B)=0.15$. Apply multiplication rule: $0.3 \times 0.5 = 0.15$.

Flashcard 22: In a uniform model with $N=20$, if $|A|=8$ and $|A\cap B|=2$, find $P(A \cap B)$.

Answer: $P(A \cap B)=\frac{2}{20}=0.10$. Two favorable outcomes out of 20 total outcomes.

Flashcard 23: In a uniform model with $\#(A)=20$ and $\#(A \cap B)=5$, find $P(B\mid A)$.

Answer: $\frac{1}{4}$. Count $B$ outcomes within $A$: $\frac{5}{20} = \frac{1}{4}$.

Flashcard 24: A fair die is rolled. Let $A=\{\text{even}\}$ and $B=\{\ge 5\}$. Find $P(B\mid A)$.

Answer: $P(B\mid A)=\frac{1}{3}$. Given even (2,4,6), only 6 satisfies $\ge 5$.

Flashcard 25: Find and correct the error: "$P(A \cap B)=P(A)+P(B\mid A)$".

Answer: Correct: $P(A \cap B)=P(A)P(B\mid A)$. Should multiply, not add, for joint probability.

Flashcard 26: If $A$ and $B$ are independent, what does the Multiplication Rule simplify to?

Answer: $P(A \cap B)=P(A)P(B)$. Independence means conditional equals marginal probability.

Flashcard 27: What does the notation $P(B\mid A)$ mean in words?

Answer: Probability that $B$ occurs given that $A$ occurred. The vertical bar means "given" or "conditional on."

Flashcard 28: Identify the required condition for $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$ to be defined.

Answer: $P(A)>0$. Division by zero is undefined when $P(A)=0$.

Flashcard 29: Identify the correct interpretation of $P(A)P(B\mid A)$ in context.

Answer: Probability $A$ occurs, then $B$ occurs given $A$ occurred. Multiplication rule gives sequential probability interpretation.

Flashcard 30: What is $P(A \cap B)$ if $P(A)=\frac{1}{4}$ and $P(B\mid A)=\frac{1}{2}$?

Answer: $\frac{1}{8}$. Apply multiplication rule: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$.

Flashcard 31: Compute $P(A\mid B)$ given $P(A \cap B)=0.18$ and $P(B)=0.6$.

Answer: $P(A\mid B)=0.3$. Divide joint probability by marginal: $0.18 \div 0.6 = 0.3$.

Flashcard 32: In a uniform model with $\#(S)=30$, $\#(A)=12$, $\#(B)=10$, $\#(A \cap B)=4$, find $P(A)P(B\mid A)$.

Answer: $\frac{2}{5}\cdot\frac{1}{3}=\frac{2}{15}$. $P(A)=\frac{12}{30}=\frac{2}{5}$, $P(B|A)=\frac{4}{12}=\frac{1}{3}$.

Flashcard 33: A deck has 52 cards. Draw 1 card. Let $A=\{\text{heart}\}$ and $B=\{\text{face card}\}$. Find $P(A \cap B)$.

Answer: $P(A \cap B)=\frac{3}{52}$. Three face cards (J,Q,K) in hearts out of 52 cards.

Flashcard 34: What is $P(B\mid A)$ if $P(A \cap B)=0.12$ and $P(A)=0.30$?

Answer: $0.40$. Divide joint probability by marginal: $0.12 \div 0.30 = 0.40$.

Flashcard 35: A fair die is rolled. Let $A=\{\text{even}\}$ and $B=\{\ge 5\}$. Find $P(A \cap B)$.

Answer: $P(A \cap B)=\frac{1}{6}$. Only outcome 6 is both even and $\ge 5$.

Flashcard 36: State the general Multiplication Rule for the intersection $P(A \cap B)$ using conditional probability.

Answer: $P(A \cap B)=P(A)P(B\mid A)$. Multiply $P(A)$ by the probability of $B$ given $A$ occurred.

Flashcard 37: A deck has 52 cards. Draw 1 card. Let $A=\{\text{heart}\}$ and $B=\{\text{face card}\}$. Find $P(B\mid A)$.

Answer: $P(B\mid A)=\frac{3}{13}$. Three face cards out of 13 hearts in the suit.