Describing Events in Sample Spaces - Statistics and Probability

1. Identify $A\cap B^c$ if $S=\{1,2,3,4,5,6\}$, $A=\{2,3,4\}$, $B=\{4,5,6\}$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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