You pick a card from a standard deck. Event \begin{align*}C\end{align*} is choosing a card that has an even number or a spade. Rewrite event \begin{align*}C\end{align*} as the combination of two events. Then, list the outcomes in event \begin{align*}C\end{align*}.
Watch This
Watch this video to learn about the concepts of unions and intersections.
https://www.youtube.com/watch?v=jAfNg3ylZAI Khan Academy: Unions and Intersections of Sets
Guidance
In the context of probability, an experiment is any occurrence that can be observed. For example, rolling a pair of dice and finding the sum of the numbers is an experiment.
An outcome is one possible result of the experiment. So, for the experiment of rolling a pair of dice and finding the sum of the numbers, one outcome is a “7” and a second outcome is an “11”.
Every experiment has one or more outcomes. The sample space, \begin{align*}S\end{align*}, of an experiment is the set of all possible outcomes. For the experiment of rolling a pair of dice and finding the sum of the numbers, the sample space is \begin{align*}S=\{ 2,3,4,5,6,7,8,9,10,11,12\}\end{align*}.
Often there is one or more outcomes that you are particularly interested in. For example, perhaps you are interested in the sum of the numbers on the dice being greater than five. The event, \begin{align*}E\end{align*}, is a subset of the sample space that includes all of the outcomes you are interested in (sometimes called the favorable outcomes). If \begin{align*}E\end{align*} is the sum of the numbers on the dice being greater than five, \begin{align*}E=\{6,7,8,9,10,11,12\}\end{align*}. There are many possible events that could be considered for any given experiment.
There are three common operations to consider with one or more events, shown in the table below. Consider the experiment of rolling a pair of dice and finding the sum of the numbers on the dice. Let \begin{align*}E\end{align*} be the event that the sum of the numbers is greater than five. \begin{align*}(E=\{6,7,8,9,10,11,12\})\end{align*}. Let \begin{align*}F\end{align*} be the event that the sum of the numbers is even \begin{align*}(F=\{2,4,6,8,10,12\})\end{align*}.
Operation |
Definition in Words |
Pair of Dice Example |
Complement of an Event \begin{align*}(E^\prime)\end{align*} |
The event that includes all outcomes in the sample space NOT in event \begin{align*}E\end{align*}. |
\begin{align*}E^\prime = \{1,2,3,4,5\}\end{align*} \begin{align*}E^\prime\end{align*} is the sum of the numbers on the dice being five or less. |
Union of Events \begin{align*}(E \cup F)\end{align*} |
The event that includes all outcomes in either event \begin{align*}E\end{align*}, event \begin{align*}F\end{align*}, or both. |
\begin{align*}E \cup F=\{2,4,6,7,8,9,10,11,12\}\end{align*} |
Intersection of Events \begin{align*}(E \cap F)\end{align*} |
The event that includes only the outcomes that occur in both event \begin{align*}E\end{align*} and event \begin{align*}F\end{align*}. |
\begin{align*}E \cap F=\{6,8,10,12\}\end{align*} |
To help visualize the way different events or combinations of events interact within a sample space, consider a Venn diagram.
The diagram above has a big rectangle for sample space \begin{align*}S\end{align*}. Within \begin{align*}S\end{align*}, the outcomes 2 through 12 appear in various places. The circle labeled \begin{align*}E\end{align*} represents event \begin{align*}E\end{align*}, and within that circle are all the outcomes in event \begin{align*}E\end{align*}. Similarly, the circle labeled \begin{align*}F\end{align*} represents event \begin{align*}F\end{align*}, and within that circle are all the outcomes in event \begin{align*}F\end{align*}. The place where the circles overlap contains the outcomes that are in both events \begin{align*}E\end{align*} and \begin{align*}F\end{align*}.
Example A
Shade the area of the diagram below that represents \begin{align*}F^\prime\end{align*}. Describe the event \begin{align*}F^\prime\end{align*}.
Solution: \begin{align*}F^\prime\end{align*} is the complement of event \begin{align*}F\end{align*}. It contains all the outcomes in the sample space that are NOT in event \begin{align*}F\end{align*}. In this case, \begin{align*}F^\prime\end{align*} is the sum of the numbers on the dice being odd.
Example B
Shade the area of the diagram below that represents \begin{align*}E \cup F\end{align*}. Then, shade the area of the diagram that represents \begin{align*}E\cap F\end{align*}. How is the union of two events different from the intersection of two events?
Solution: \begin{align*}E \cup F\end{align*} is the union of events \begin{align*}E\end{align*} and \begin{align*}F\end{align*}. It contains all the outcomes that are in event \begin{align*}E\end{align*}, event \begin{align*}F\end{align*}, or both events E and \begin{align*}F\end{align*}. The symbol \begin{align*}\cup\end{align*} can be thought of as “or”, but remember that it is not exclusive or, since it includes outcomes that are in both events. \begin{align*}E \cup F\end{align*} is shown below.
\begin{align*}E \cap F\end{align*} is the intersection of events \begin{align*}E\end{align*} and \begin{align*}F\end{align*}. It contains all the outcomes that are in both events \begin{align*}E\end{align*} and \begin{align*}F\end{align*}. The symbol \begin{align*}\cap\end{align*} can be thought of as “and”, since it includes only the outcomes that are in both events. \begin{align*}E \cap F\end{align*} is shown below.
Notice that \begin{align*}E \cup F\end{align*} will always contain the same outcomes as \begin{align*}E \cap F\end{align*}, plus more outcomes (usually). \begin{align*}E \cup F\end{align*} could never contain less outcomes than \begin{align*}E \cap F\end{align*}.
Example C
Consider the experiment of tossing three coins and recording the sequence of heads and tails. Let \begin{align*}A\end{align*} be the event that there are exactly two heads. Let \begin{align*}B\end{align*} be the event that there are exactly two tails.
a) Find the sample space for the experiment.
b) List the outcomes in event \begin{align*}A\end{align*}.
c) List the outcomes in event \begin{align*}B\end{align*}.
d) Create a diagram that shows the sample space, events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, and all of the outcomes.
Solution: Typically when working with experiments having to do with coins, \begin{align*}H\end{align*} represents getting “heads” and \begin{align*}T\end{align*} represents getting “tails”.
a) The sample space is: \begin{align*}S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}\end{align*}
b) \begin{align*}A=\{HHT,HTH,THH\}\end{align*}
c) \begin{align*}B=\{HTT,THT,TTH\}\end{align*}
d) Notice that this time there are no outcomes that are in both \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, so the circles don't overlap.
Concept Problem Revisited
There are 52 cards in a standard deck of cards. These 52 cards are organized by suit:
Clubs: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King
Diamonds: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King
Hearts: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King
Spades: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King
Event \begin{align*}C\end{align*} is choosing a card that has an even number or a spade on it. Because the word “or” is used, you know this is really the union of two other events. Let \begin{align*}D\end{align*} be choosing an even number and let \begin{align*}E\end{align*} be choosing a spade. Then \begin{align*}C=D \cup E\end{align*}.
\begin{align*}D=\end{align*} {2 of clubs, 4 of clubs, 6 of clubs, 8 of clubs, 10 of clubs, 2 of diamonds, 4 of diamonds, 6 of diamonds, 8 of diamonds, 10 of diamonds, 2 of hearts, 4 of hearts, 6 of hearts, 8 of hearts, 10 of hearts, 2 of spades, 4 of spades, 6 of spades, 8 of spades, 10 of spades}
\begin{align*}E=\end{align*} {Ace of spades, 2 of spades, 3 of spades, 4 of spades, 5 of spades, 6 of spades, 7 of spades, 8 of spades, 9 of spades, 10 of spades, Jack of spades, Queen of spades, King of spades}
\begin{align*}C=D \cup E=\end{align*} {Ace of spades, 2 of spades, 3 of spades, 4 of spades, 5 of spades, 6 of spades, 7 of spades, 8 of spades, 9 of spades, 10 of spades, Jack of spades, Queen of spades, King of spades, 2 of clubs, 4 of clubs, 6 of clubs, 8 of clubs, 10 of clubs, 2 of diamonds, 4 of diamonds, 6 of diamonds, 8 of diamonds, 10 of diamonds, 2 of hearts, 4 of hearts, 6 of hearts, 8 of hearts, 10 of hearts}
Vocabulary
An experiment is an occurrence with a result that can be observed.
An outcome of an experiment is one possible result of the experiment.
The sample space for an experiment is the set of all possible outcomes of the experiment.
An event for an experiment is a subset of the sample space containing outcomes that you are interested in (sometimes called favorable outcomes).
The complement of an event is the event that includes all outcomes in the sample space not in the original event. The symbol for complement is ′.
The union of two events is the event that includes all outcomes that are in either or both of the original events. The symbol for union is \begin{align*}\cup\end{align*}.
The intersection of two events is the event that includes all outcomes that are in both of the original events. The symbol for intersection is \begin{align*}\cap\end{align*}.
A Venn diagram is a way to visualize sample spaces, events, and outcomes.
Guided Practice
Consider the experiment from Example C of tossing three coins and recording the sequence of heads and tails. The diagram below represents the sample space and two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.
1. Describe \begin{align*}A^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
2. Describe \begin{align*}(A \cup B)\end{align*} in words and with the diagram. What outcomes are in this event?
3. Describe \begin{align*}(A \cup B)^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
4. Describe \begin{align*}(A \cap B)^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
Answers:
1. \begin{align*}A^\prime\end{align*} is the event of not getting exactly two heads. \begin{align*}A^\prime=\{TTH,THT,HTT,HHH,TTT\}\end{align*} On the diagram, it is everything in the rectangle except circle \begin{align*}A\end{align*}.
2. \begin{align*}(A \cup B)\end{align*} is all of the outcomes in event \begin{align*}A\end{align*}, event \begin{align*}B\end{align*}, or both. Note that for this experiment, there are no events in both \begin{align*}A\end{align*} and \begin{align*}B\end{align*}. \begin{align*}A \cup B=\{HHT, HTH, THH, TTH, THT, HTT\}\end{align*}. In the diagram, only circles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are shaded.
3. \begin{align*}(A \cup B)^\prime\end{align*} is all of the outcomes not in \begin{align*}(A \cup B)\end{align*}. This means it is all of the outcomes that are in neither events \begin{align*}A\end{align*} nor \begin{align*}B\end{align*}. \begin{align*}(A \cup B)^\prime=\{HHH,TTT\}\end{align*}. In the diagram, the opposite part of the rectangle is shaded compared with #2.
4. \begin{align*}(A \cap B)^\prime\end{align*} is all of the outcomes not in \begin{align*}(A \cap B)\end{align*}. Remember that \begin{align*}A \cap B\end{align*} is all of the outcomes in both events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}; however, in this experiment the two events don't overlap (they are disjoint).
\begin{align*}A \cap B=\{\}\end{align*}, the empty set. This means that \begin{align*}(A \cap B)^\prime\end{align*} must be the whole sample space, since it has to include all outcomes in the sample space not in \begin{align*}A \cap B\end{align*}.
\begin{align*}(A \cap B)^\prime=\{HHH,HHT,HTH,THH,TTH,THT,HTT,TTT\}\end{align*}. In the diagram, everything in the rectangle is shaded.
Practice
Consider the experiment of spinning the spinner below twice and recording the sequence of results. Let \begin{align*}F\end{align*} be the event that the same color comes up twice. Let \begin{align*}H\end{align*} be the event that there is at least one red.
1. Find the sample space for the experiment.
2. List the outcomes in event \begin{align*}F\end{align*} and the outcomes in event \begin{align*}H\end{align*}.
3. Create a Venn diagram that shows the relationships between the outcomes in \begin{align*}F\end{align*}, \begin{align*}H\end{align*}, and the sample space.
4. Describe \begin{align*}F^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
5. Describe the event “getting two reds” with symbols and with the diagram. What outcomes are in this event?
6. Describe \begin{align*}(F \cup H)^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
Consider the experiment of rolling a pair of dice and finding the sum of the numbers on the dice. Let \begin{align*}J\end{align*} be the event that the sum is less than 4. Let \begin{align*}K\end{align*} be the event that the sum is an odd number.
7. Find the sample space for the experiment.
8. List the outcomes in event \begin{align*}J\end{align*} and the outcomes in event \begin{align*}K\end{align*}.
9. Create a Venn diagram that shows the relationships between the outcomes in \begin{align*}J, K\end{align*}, and the sample space.
10. Describe \begin{align*}K^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
11. Describe the event “getting an even number less than 4” with symbols and with the diagram. What outcomes are in this event?
12. Describe \begin{align*}(J \cap K^\prime)^\prime\end{align*} in words and with the diagram. What outcomes are in this event?
13. In this experiment, are you just as likely to get a sum of 2 as a sum of 7? Explain.
14. Compare and contrast unions of events with intersections of events.
15. Consider some experiment with event \begin{align*}E\end{align*}. Describe \begin{align*}E \cup E^\prime\end{align*}. Describe \begin{align*}E \cap E^\prime\end{align*}.