Statistics Flashcards: Describing Events In Sample Spaces

What this deck covers

This deck focuses on Describing Events In Sample Spaces, 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 $A\cap B^c$ if $S=\{1,2,3,4,5,6\}$, $A=\{2,3,4\}$, $B=\{4,5,6\}$.

Answer: $A\cap B^c=\{2,3\}$. $B^c=\{1,2,3\}$, so $A\cap B^c$ finds common elements.

Flashcard 2: Identify the event: roll one die, $A=$ "odd," $B=$ "prime." What is $A \cap B$?

Answer: $A \cap B=\{3,5\}$. 3 and 5 are both odd and prime numbers.

Flashcard 3: What does it mean if $A \subseteq B$ for events $A$ and $B$?

Answer: Every outcome in $A$ is also in $B$. Event $A$ occurring guarantees event $B$ occurs.

Flashcard 4: What does the union $A \cup B$ represent in words?

Answer: Outcomes in $A$ or $B$ (or both). The "or" operation includes outcomes from either set.

Flashcard 5: Find $A\cup B$ if $A=\{1,2,3\}$ and $B=\{3,4,5\}$.

Answer: $A\cup B=\{1,2,3,4,5\}$. Union combines all elements from both sets.

Flashcard 6: What does the complement $A^c$ represent, relative to sample space $S$?

Answer: $A^c=\{\omega\in S: \omega\notin A\}$. Contains all outcomes in $S$ that are not in $A$.

Flashcard 7: What is the definition of a sample space $S$ in probability?

Answer: The set $S$ of all possible outcomes of a random process. Contains every possible result that could occur.

Flashcard 8: Find $A\cap B$ if $A=\{1,2,3\}$ and $B=\{3,4,5\}$.

Answer: $A\cap B=\{3\}$. Only 3 appears in both sets.

Flashcard 9: Identify the event for "an even number" when $S=\{1,2,3,4,5,6\}$.

Answer: $\{2,4,6\}$. Even numbers in the sample space are 2, 4, and 6.

Flashcard 10: Find $A^c$ if $S=\{1,2,3,4,5,6\}$ and $A=\{1,3,5\}$.

Answer: $A^c=\{2,4,6\}$. Complement contains all elements not in $A$.

Flashcard 11: What is De Morgan's law for the complement of a union?

Answer: $(A\cup B)^c=A^c\cap B^c$. Not in union means not in either set.

Flashcard 12: Identify the event "die is even or coin is $H$" for $S=\{(1,H),\ldots,(6,T)\}$.

Answer: $\{(2,H),(2,T),(4,H),(4,T),(6,H),(6,T),(1,H),(3,H),(5,H)\}$. Union of even die outcomes and heads outcomes.

Flashcard 13: What is an event $A$ in probability, stated using a sample space $S$?

Answer: An event $A$ is any subset of $S$. Any collection of outcomes from the sample space forms an event.

Flashcard 14: Which expression represents "not ($A$ and $B$)" using set notation?

Answer: $(A\cap B)^c$. Complement of intersection negates the "and" expression.

Flashcard 15: What is the definition of an event $A$ in terms of a sample space $S$?

Answer: An event is any subset $A \subseteq S$. Events are collections of outcomes from the sample space.

Flashcard 16: What does the intersection $A \cap B$ represent in words?

Answer: Outcomes in both $A$ and $B$. The "and" operation requires membership in both sets.

Flashcard 17: Identify the event: flip two coins, $S=\{HH,HT,TH,TT\}$, $A=$ "exactly one head." What is $A$?

Answer: $A=\{HT,TH\}$. HT and TH each have exactly one head.

Flashcard 18: What does the notation $\omega \in A$ mean for an event $A$?

Answer: Outcome $\omega$ is in event $A$ (so event $A$ occurs). The outcome satisfies the conditions defining event $A$.

Flashcard 19: What does the intersection $A\cap B$ represent in words for events $A$ and $B$?

Answer: $A\cap B$ means both $A$ and $B$. Intersection includes only outcomes in both events.

Flashcard 20: What does the set difference $A \setminus B$ represent in words?

Answer: Outcomes in $A$ but not in $B$. Removes outcomes that belong to $B$ from set $A$.

Flashcard 21: What does it mean if two events are disjoint, written $A \cap B = \varnothing$?

Answer: They share no outcomes; they cannot occur together. Disjoint events have no overlap in their outcomes.

Flashcard 22: What is the sample space $S$ for one roll of a standard six-sided die?

Answer: $S=\{1,2,3,4,5,6\}$. Lists all possible outcomes when rolling a standard die.

Flashcard 23: Find $A \cap B$: roll one die, $A=\{1,2,3\}$ and $B=\{3,4\}$. What is $A \cap B$?

Answer: $A \cap B=\{3\}$. Only outcome 3 appears in both sets.

Flashcard 24: Find $A \setminus B$: roll one die, $A=\{1,2,3\}$ and $B=\{3,4\}$. What is $A \setminus B$?

Answer: $A \setminus B=\{1,2\}$. Removes the shared outcome 3 from set $A$.

Flashcard 25: What is De Morgan's law for the complement of an intersection?

Answer: $(A\cap B)^c=A^c\cup B^c$. Not in both means not in at least one.

Flashcard 26: Identify the event: roll one die, $S=\{1,2,3,4,5,6\}$, $A=$ "greater than $4$." What is $A$?

Answer: $A=\{5,6\}$. Only 5 and 6 exceed 4 on a standard die.

Flashcard 27: Find $A \cup B$: roll one die, $A=\{1,2,3\}$ and $B=\{3,4\}$. What is $A \cup B$?

Answer: $A \cup B=\{1,2,3,4\}$. Union combines all outcomes from both sets.

Flashcard 28: Find $A^c$: roll one die, $S=\{1,2,3,4,5,6\}$, $A=\{1,2,3\}$. What is $A^c$?

Answer: $A^c=\{4,5,6\}$. Complement includes all outcomes not in $A$.

Flashcard 29: Identify the event $A \cap B$: two coins, $A=$ "first coin is $H$," $B=$ "second coin is $H$." What is $A \cap B$?

Answer: $A \cap B=\{HH\}$. Both coins must show heads for intersection.

Flashcard 30: Identify the event $A \cup B$: two coins, $A=$ "first coin is $H$," $B=$ "second coin is $H$." What is $A \cup B$?

Answer: $A \cup B=\{HH,HT,TH\}$. At least one coin shows heads in these outcomes.

Flashcard 31: Identify the event: roll one die, $S=\{1,2,3,4,5,6\}$, $A=$ "even." What is $A$?

Answer: $A=\{2,4,6\}$. Even numbers on a die are 2, 4, and 6.

Flashcard 32: What is the complement of an event $A$ in a sample space $S$?

Answer: $A^c = S \setminus A$. Contains all outcomes not in $A$.

Flashcard 33: What is the sample space $S$ for rolling one die and flipping one coin (die first)?

Answer: $S=\{(1,H),\ldots,(6,H),(1,T),\ldots,(6,T)\}$. Ordered pairs of (die result, coin result) for all combinations.

Flashcard 34: Which expression represents "not ($A$ or $B$)" using set notation?

Answer: $(A\cup B)^c$. Complement of the union negates the entire "or" expression.

Flashcard 35: Identify the event for "a number greater than $4$" when $S=\{1,2,3,4,5,6\}$.

Answer: $\{5,6\}$. Only 5 and 6 are greater than 4 in this sample space.

Flashcard 36: What does the union $A\cup B$ represent in words for events $A$ and $B$?

Answer: $A\cup B$ means $A$ or $B$ (or both). Union includes outcomes in either event.

Flashcard 37: Identify the event "exactly one head" for two coin flips with $S=\{HH,HT,TH,TT\}$.

Answer: $\{HT,TH\}$. HT and TH each have exactly one H.

Flashcard 38: Which expression represents "$A$ and not $B$" using set notation?

Answer: $A\cap B^c$. Intersection of $A$ with the complement of $B$.

Flashcard 39: What does the notation $\omega \in S$ mean?

Answer: $\omega$ is an outcome in the sample space $S$. Indicates that $\omega$ is one of the possible outcomes.

Flashcard 40: What is the sample space $S$ for flipping a coin twice, using $H$ and $T$?

Answer: $S=\{HH,HT,TH,TT\}$. All possible sequences of two coin flips.