Describing Events in Sample Spaces - Statistics
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What is De Morgan's law for the complement of a union?
What is De Morgan's law for the complement of a union?
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$(A\cup B)^c=A^c\cap B^c$. Not in union means not in either set.
$(A\cup B)^c=A^c\cap B^c$. Not in union means not in either set.
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What is De Morgan's law for the complement of an intersection?
What is De Morgan's law for the complement of an intersection?
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$(A\cap B)^c=A^c\cup B^c$. Not in both means not in at least one.
$(A\cap B)^c=A^c\cup B^c$. Not in both means not in at least one.
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Identify the event for “an even number” when $S={1,2,3,4,5,6}$.
Identify the event for “an even number” when $S={1,2,3,4,5,6}$.
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${2,4,6}$. Even numbers in the sample space are 2, 4, and 6.
${2,4,6}$. Even numbers in the sample space are 2, 4, and 6.
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Identify the event $A \cap B$: two coins, $A=$ “first coin is $H$,” $B=$ “second coin is $H$.” What is $A \cap B$?
Identify the event $A \cap B$: two coins, $A=$ “first coin is $H$,” $B=$ “second coin is $H$.” What is $A \cap B$?
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$A \cap B={HH}$. Both coins must show heads for intersection.
$A \cap B={HH}$. Both coins must show heads for intersection.
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What does the notation $\omega \in S$ mean?
What does the notation $\omega \in S$ mean?
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$\omega$ is an outcome in the sample space $S$. Indicates that $\omega$ is one of the possible outcomes.
$\omega$ is an outcome in the sample space $S$. Indicates that $\omega$ is one of the possible outcomes.
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Identify the event for “a number greater than $4$” when $S={1,2,3,4,5,6}$.
Identify the event for “a number greater than $4$” when $S={1,2,3,4,5,6}$.
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${5,6}$. Only 5 and 6 are greater than 4 in this sample space.
${5,6}$. Only 5 and 6 are greater than 4 in this sample space.
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Find $A^c$ if $S={1,2,3,4,5,6}$ and $A={1,3,5}$.
Find $A^c$ if $S={1,2,3,4,5,6}$ and $A={1,3,5}$.
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$A^c={2,4,6}$. Complement contains all elements not in $A$.
$A^c={2,4,6}$. Complement contains all elements not in $A$.
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Find $A\cup B$ if $A={1,2,3}$ and $B={3,4,5}$.
Find $A\cup B$ if $A={1,2,3}$ and $B={3,4,5}$.
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$A\cup B={1,2,3,4,5}$. Union combines all elements from both sets.
$A\cup B={1,2,3,4,5}$. Union combines all elements from both sets.
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Find $A\cap B$ if $A={1,2,3}$ and $B={3,4,5}$.
Find $A\cap B$ if $A={1,2,3}$ and $B={3,4,5}$.
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$A\cap B={3}$. Only 3 appears in both sets.
$A\cap B={3}$. Only 3 appears in both sets.
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Identify $A\cap B^c$ if $S={1,2,3,4,5,6}$, $A={2,3,4}$, $B={4,5,6}$.
Identify $A\cap B^c$ if $S={1,2,3,4,5,6}$, $A={2,3,4}$, $B={4,5,6}$.
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$A\cap B^c={2,3}$. $B^c={1,2,3}$, so $A\cap B^c$ finds common elements.
$A\cap B^c={2,3}$. $B^c={1,2,3}$, so $A\cap B^c$ finds common elements.
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What is the sample space $S$ for flipping a coin twice, using $H$ and $T$?
What is the sample space $S$ for flipping a coin twice, using $H$ and $T$?
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$S={HH,HT,TH,TT}$. All possible sequences of two coin flips.
$S={HH,HT,TH,TT}$. All possible sequences of two coin flips.
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Identify the event “exactly one head” for two coin flips with $S={HH,HT,TH,TT}$.
Identify the event “exactly one head” for two coin flips with $S={HH,HT,TH,TT}$.
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${HT,TH}$. HT and TH each have exactly one H.
${HT,TH}$. HT and TH each have exactly one H.
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What does the complement $A^c$ represent, relative to sample space $S$?
What does the complement $A^c$ represent, relative to sample space $S$?
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$A^c={\omega\in S: \omega\notin A}$. Contains all outcomes in $S$ that are not in $A$.
$A^c={\omega\in S: \omega\notin A}$. Contains all outcomes in $S$ that are not in $A$.
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What is the sample space $S$ for rolling one die and flipping one coin (die first)?
What is the sample space $S$ for rolling one die and flipping one coin (die first)?
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$S={(1,H),\ldots,(6,H),(1,T),\ldots,(6,T)}$. Ordered pairs of (die result, coin result) for all combinations.
$S={(1,H),\ldots,(6,H),(1,T),\ldots,(6,T)}$. Ordered pairs of (die result, coin result) for all combinations.
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Identify the event “die is even or coin is $H$” for $S={(1,H),\ldots,(6,T)}$.
Identify the event “die is even or coin is $H$” for $S={(1,H),\ldots,(6,T)}$.
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${(2,H),(2,T),(4,H),(4,T),(6,H),(6,T),(1,H),(3,H),(5,H)}$. Union of even die outcomes and heads outcomes.
${(2,H),(2,T),(4,H),(4,T),(6,H),(6,T),(1,H),(3,H),(5,H)}$. Union of even die outcomes and heads outcomes.
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What does the intersection $A\cap B$ represent in words for events $A$ and $B$?
What does the intersection $A\cap B$ represent in words for events $A$ and $B$?
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$A\cap B$ means both $A$ and $B$. Intersection includes only outcomes in both events.
$A\cap B$ means both $A$ and $B$. Intersection includes only outcomes in both events.
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What does the union $A\cup B$ represent in words for events $A$ and $B$?
What does the union $A\cup B$ represent in words for events $A$ and $B$?
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$A\cup B$ means $A$ or $B$ (or both). Union includes outcomes in either event.
$A\cup B$ means $A$ or $B$ (or both). Union includes outcomes in either event.
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What is an event $A$ in probability, stated using a sample space $S$?
What is an event $A$ in probability, stated using a sample space $S$?
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An event $A$ is any subset of $S$. Any collection of outcomes from the sample space forms an event.
An event $A$ is any subset of $S$. Any collection of outcomes from the sample space forms an event.
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What is the sample space $S$ for one roll of a standard six-sided die?
What is the sample space $S$ for one roll of a standard six-sided die?
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$S={1,2,3,4,5,6}$. Lists all possible outcomes when rolling a standard die.
$S={1,2,3,4,5,6}$. Lists all possible outcomes when rolling a standard die.
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Which expression represents “$A$ and not $B$” using set notation?
Which expression represents “$A$ and not $B$” using set notation?
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$A\cap B^c$. Intersection of $A$ with the complement of $B$.
$A\cap B^c$. Intersection of $A$ with the complement of $B$.
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Which expression represents “not ($A$ or $B$)” using set notation?
Which expression represents “not ($A$ or $B$)” using set notation?
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$(A\cup B)^c$. Complement of the union negates the entire "or" expression.
$(A\cup B)^c$. Complement of the union negates the entire "or" expression.
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Which expression represents “not ($A$ and $B$)” using set notation?
Which expression represents “not ($A$ and $B$)” using set notation?
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$(A\cap B)^c$. Complement of intersection negates the "and" expression.
$(A\cap B)^c$. Complement of intersection negates the "and" expression.
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What does it mean if $A \subseteq B$ for events $A$ and $B$?
What does it mean if $A \subseteq B$ for events $A$ and $B$?
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Every outcome in $A$ is also in $B$. Event $A$ occurring guarantees event $B$ occurs.
Every outcome in $A$ is also in $B$. Event $A$ occurring guarantees event $B$ occurs.
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Identify the event: roll one die, $A=$ “odd,” $B=$ “prime.” What is $A \cap B$?
Identify the event: roll one die, $A=$ “odd,” $B=$ “prime.” What is $A \cap B$?
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$A \cap B={3,5}$. 3 and 5 are both odd and prime numbers.
$A \cap B={3,5}$. 3 and 5 are both odd and prime numbers.
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Identify the event $A \cup B$: two coins, $A=$ “first coin is $H$,” $B=$ “second coin is $H$.” What is $A \cup B$?
Identify the event $A \cup B$: two coins, $A=$ “first coin is $H$,” $B=$ “second coin is $H$.” What is $A \cup B$?
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$A \cup B={HH,HT,TH}$. At least one coin shows heads in these outcomes.
$A \cup B={HH,HT,TH}$. At least one coin shows heads in these outcomes.
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Identify the event: flip two coins, $S={HH,HT,TH,TT}$, $A=$ “exactly one head.” What is $A$?
Identify the event: flip two coins, $S={HH,HT,TH,TT}$, $A=$ “exactly one head.” What is $A$?
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$A={HT,TH}$. HT and TH each have exactly one head.
$A={HT,TH}$. HT and TH each have exactly one head.
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Find $A \setminus B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \setminus B$?
Find $A \setminus B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \setminus B$?
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$A \setminus B={1,2}$. Removes the shared outcome 3 from set $A$.
$A \setminus B={1,2}$. Removes the shared outcome 3 from set $A$.
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Find $A \cap B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \cap B$?
Find $A \cap B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \cap B$?
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$A \cap B={3}$. Only outcome 3 appears in both sets.
$A \cap B={3}$. Only outcome 3 appears in both sets.
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Find $A \cup B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \cup B$?
Find $A \cup B$: roll one die, $A={1,2,3}$ and $B={3,4}$. What is $A \cup B$?
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$A \cup B={1,2,3,4}$. Union combines all outcomes from both sets.
$A \cup B={1,2,3,4}$. Union combines all outcomes from both sets.
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Find $A^c$: roll one die, $S={1,2,3,4,5,6}$, $A={1,2,3}$. What is $A^c$?
Find $A^c$: roll one die, $S={1,2,3,4,5,6}$, $A={1,2,3}$. What is $A^c$?
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$A^c={4,5,6}$. Complement includes all outcomes not in $A$.
$A^c={4,5,6}$. Complement includes all outcomes not in $A$.
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