Statistics Flashcards: Understanding Independent Events

What this deck covers

This deck focuses on Understanding Independent Events, 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: Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{5}$.

Answer: Independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5}$, which equals $P(A \cap B)$.

Flashcard 2: Which statement is always true about independence: does $A$ being independent of $B$ imply $B$ independent of $A$?

Answer: Yes, independence is symmetric. If $P(A\cap B)=P(A)P(B)$, then both directions hold.

Flashcard 3: Identify whether $A$ and $B$ are independent if $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{6}$.

Answer: Independent. $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$, which equals $P(A\cap B)$.

Flashcard 4: What equation tests independence using $P(A\mid B)$ when $P(B)>0$?

Answer: $P(A\mid B)=P(A)$. If $B$ doesn't affect $A$'s probability, they're independent.

Flashcard 5: What is the independence condition using the product rule for events $A$ and $B$?

Answer: $P(A \cap B)=P(A)P(B)$. Independent events satisfy this multiplication rule.

Flashcard 6: Find $P(B)$ if $A$ and $B$ are independent with $P(A)=0.25$ and $P(A\cap B)=0.05$.

Answer: $0.2$. Solve: $0.25 \times P(B) = 0.05$, so $P(B) = 0.2$.

Flashcard 7: What is the independence test written using conditional probability $P(B\mid A)$?

Answer: $P(B\mid A)=P(B)$. For independent events, knowing A occurred doesn't change the probability of B.

Flashcard 8: Identify whether $A$ and $B$ are independent from this table: $P(A)=0.4$, $P(B)=0.3$, $P(A\cap B)=0.10$.

Answer: Not independent. $0.4 \times 0.3 = 0.12 \neq 0.10$.

Flashcard 9: Find and correct the mistake: claiming independence because $P(A\cap B)=P(A)+P(B)$.

Answer: Correct test: $P(A\cap B)=P(A)P(B)$. Addition rule is for disjoint events, not independence.

Flashcard 10: Find $P(B)$ if $A$ and $B$ are independent, $P(A)=0.25$, and $P(A\cap B)=0.05$.

Answer: $0.20$. Solve $0.25 \times P(B) = 0.05$ to get $P(B) = 0.20$.

Flashcard 11: Identify whether $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

Answer: Independent. $0.4 \times 0.5 = 0.2$, which equals $P(A\cap B)$.

Flashcard 12: Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{4}$.

Answer: Not independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5} \neq \frac{1}{4}$.

Flashcard 13: Identify whether $A$ and $B$ are independent if $P(A)=0.3$, $P(B)=0.6$, and $P(A\cap B)=0.25$.

Answer: Not independent. $0.3 \times 0.6 = 0.18 \neq 0.25$.

Flashcard 14: What is the defining equation for independence using the intersection probability?

Answer: $P(A \cap B)=P(A)P(B)$. Events are independent when their joint probability equals the product of individual probabilities.

Flashcard 15: Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=\frac{3}{4}$ and $P(B)=\frac{2}{3}$.

Answer: $\frac{1}{2}$. Multiply: $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$.

Flashcard 16: Find $P(A)$ if $A$ and $B$ are independent, $P(B)=0.8$, and $P(A\cap B)=0.12$.

Answer: $0.15$. Solve $P(A) \times 0.8 = 0.12$ to get $P(A) = 0.15$.

Flashcard 17: Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.12$.

Answer: $0.12$. For independent events, $P(B \mid A) = P(B)$.

Flashcard 18: What is the symmetric conditional form of independence between events $A$ and $B$?

Answer: $P(A\mid B)=P(A)$ and $P(B\mid A)=P(B)$. Independence means each event's probability is unchanged by the other.

Flashcard 19: Identify whether $A$ and $B$ are independent if $P(A)=0.6$, $P(B)=0.3$, and $P(A\cap B)=0.25$.

Answer: Not independent. Check: $0.6 \times 0.3 = 0.18 \neq 0.25$.

Flashcard 20: Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.

Answer: $0.35$. For independent events, $P(A \mid B) = P(A)$.

Flashcard 21: Identify whether $A$ and $B$ are independent if $P(A)=0.5$, $P(B)=0.4$, and $P(A\mid B)=0.5$.

Answer: Independent. Since $P(A \mid B) = P(A) = 0.5$, they are independent.

Flashcard 22: Which formula gives conditional probability $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?

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

Flashcard 23: Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.65$.

Answer: $0.65$. For independent events, $P(A\mid B) = P(A)$.

Flashcard 24: Which statement correctly describes independence: $P(A\cap B)>P(A)P(B)$, $=$, or $<$?

Answer: $P(A\cap B)=P(A)P(B)$. Independence requires equality, not greater than or less than.

Flashcard 25: Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.5$ with $P(B)>0$.

Answer: Independent. $P(A\mid B) = P(A)$ confirms independence.

Flashcard 26: Which statement is true if $A$ and $B$ are independent: $P(A\cap B)$ equals what expression?

Answer: $P(A\cap B)=P(A)P(B)$. This is the defining property of independent events.

Flashcard 27: What is $P(A \cap B)$ if $A$ and $B$ are independent and you know $P(A)$ and $P(B)$?

Answer: $P(A \cap B)=P(A)P(B)$. For independent events, multiply their individual probabilities.

Flashcard 28: Find $P(A)$ if $A$ and $B$ are independent with $P(B)=0.8$ and $P(A\cap B)=0.24$.

Answer: $0.3$. Solve: $P(A) \times 0.8 = 0.24$, so $P(A) = 0.3$.

Flashcard 29: Which probability must equal $P(A)P(B)$ for events $A$ and $B$ to be independent?

Answer: $P(A \cap B)$. The intersection probability must equal the product for independence.

Flashcard 30: What equation tests independence using $P(B\mid A)$ when $P(A)>0$?

Answer: $P(B\mid A)=P(B)$. If $A$ doesn't affect $B$'s probability, they're independent.

Flashcard 31: Identify whether $A$ and $B$ are independent if $P(A)=0.5$, $P(B)=0.4$, and $P(A\mid B)=0.6$.

Answer: Not independent. Since $P(A \mid B) = 0.6 \neq 0.5 = P(A)$, they're not independent.

Flashcard 32: What is the equivalent independence condition using $P(A\cap B)$ and $P(B)$ when $P(B)>0$?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=P(A)$. Rearranging the conditional probability formula shows independence.

Flashcard 33: Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.7$ and $P(B)=0.2$.

Answer: $0.14$. Multiply: $0.7 \times 0.2 = 0.14$.

Flashcard 34: What is the independence test written using conditional probability $P(A\mid B)$?

Answer: $P(A\mid B)=P(A)$. For independent events, knowing B occurred doesn't change the probability of A.

Flashcard 35: Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.7$, and $P(B)=0.2$.

Answer: $0.14$. For independent events, multiply: $0.7 \times 0.2 = 0.14$.

Flashcard 36: Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.7$ with $P(B)>0$.

Answer: Not independent. $P(A\mid B) \neq P(A)$ means not independent.

Flashcard 37: Which formula gives conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$?

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