Sample Spaces and Events
Sample Spaces and Events
Basic Concepts in Probability:
Experiment: any action or process whose outcome is subject to uncertainty e.g. tossing a coin 3 times, testing the pH value of some reagent, counting the number of customers visiting a store in one day, etc.
Sample Space: the set of all possible outcomes of an experiment, usually denoted by S e.g. for the above 3 examples, the sample spaces are { TTT, TTH,
THH, THT, HHH, HHT, HTH, HTT } , [0,14] and { 0, 1, 2, . . . , N,
. . .
} , respectively.
Liang Zhang (UofU)
June 10, 2008 1 / 10
Sample Spaces and Events
Basic Concepts in Probability:
Event: any colletcion (subset) of outcomes contained in the sample space S .
An event is simle if it consists of exactly one outcome and compound if it consists of more than one outcome.
e.g. for the coin tossing example: { all the outcomes such that the first result is Head } , i.e.
{ HHT, HTH, HTT, HHH } , is an event and this is a compoud event;
{ all the outcomes which have 3 consecutive Head } , i.e.
{ HHH } , is also an event, while this is a single event.
Liang Zhang (UofU)
June 10, 2008 2 / 10
Sample Spaces and Events
Examples:
For the pH value testing example:
{ pH value is less than 7.0
} , i.e. [0, 7.0), is an event, and it is compound;
{ pH value is between 2.0 and 3.0
} , i.e. [2.0, 3.0], is another event, and it is also compound.
For the customers’ visiting investigation example:
{ the number of cumstomers visited in one day is less than 100 } , i.e.
{ 1, 2,
3, . . . , 98, 99 } , is an event, and it is compound;
{ the number of cumstomers visited in one day is more than 200 } , i.e.
{ 201, 202, . . .
} is also an event and it is compound.
Liang Zhang (UofU)
June 10, 2008 3 / 10
Sample Spaces and Events
Another Coin Tossing Example:
This time our experiment is tossing a coin many times until we get our first Head.
Then the sample space would be { 1, 2, 3, 4, 5, . . .
} and the corresponding outcomes are { H, TH, TTH, TTTH, TTTTH, · · · } .
Here are some events:
{ we tossed at most 3 times } , i.e.
{ 1, 2, 3 } or { H, TH, TTH } .
{ we totally tossed an even number of times } , i.e.
{ 2, 4, 6, . . .
} or { TH,
TTTH, TTTTTH, · · · } .
Both events are compound.
Liang Zhang (UofU)
June 10, 2008 4 / 10
Sample Spaces and Events
Basic Set Theory
Complement: the complement of an event A denoted by A’ is the set of all outcomes in S that are not contained in A.
e.g. for our first coin tossing example, if
A = { the first outcome is Head } = { HHH, HHT, HTH, HTT } , then
A’ = { the first outcome is not Head, i.e. Tail } = { TTT, TTH, THT,
THH } for the pH value testing example, if
A = { the pH value of the reagent is below 7.0
} , then
A’ = { the the pH value of the reagent is above 7.0
}
Liang Zhang (UofU)
June 10, 2008 5 / 10
Sample Spaces and Events
Basic Set Theory
Union: the union of two events A and B, is the event consisting of all outcomes that are eigther in A or in B or in both events — that is, all outcomes in at least one of the events, denoted by A ∪ B e.g. for the coin tossing example, if
A = { the first outcome is Head } = { HHH, HHT, HTH, HTT } , and
B = { the last outcome is Head } = { HHH, TTH, HTH, THH } , then
A ∪ B = { the first or the last outcomem is Head }
= { HHH , HHT , HTH , HTT , TTH , THH }
Liang Zhang (UofU)
June 10, 2008 6 / 10
Sample Spaces and Events
Basic Set Theory
Intersection: the intersection of two events A and B, is the event consisting of all outcomes that are both in A and in B , denoted by
A ∩ B e.g. for the coin tossing example, if
A = { the first outcome is Head } = { HHH, HHT, HTH, HTT } , and
B = { the last outcome is Head } = { HHH, TTH, HTH, THH } , then
A ∩ B = { the first and the last outcomem is Head }
= { HHH , HTH }
Liang Zhang (UofU)
June 10, 2008 7 / 10
Sample Spaces and Events
Basic Set Theory
Null Event: the event consistion of no outcomes, denoted by ∅ e.g. the event { the first outcome is neither Head nor Tail } for the coin tossing experiment is a null event .
Mutually Exclusive: if two events A and B satisfy A ∩ B = ∅ , then A and B are said to be mutually exclusive or mutually disjoint .
e.g. for the coin tossing example, if
A = { the first outcome is Head } = { HHH, HHT, HTH, HTT } , and
B = { the first outcome is Tail } = { THH, TTH, TTT, THT } , then
A ∩ B = { the first outcomem is Head and Tail }
= ∅
So A and B are mutually disjoint .
Liang Zhang (UofU)
June 10, 2008 8 / 10
Sample Spaces and Events
Remark:
1. The union and intersection operation can be extended to more than two events.
e.g. for any three events A, B and C, the event A ∪ B ∪ C is the set of all outcomes contained in at least one of the three events;
Similarly, A ∩ B ∩ C is the set of all outcomes contained in all three events.
2. Given n events A
1
, A
2
, . . . , A n
. They are said to be mutually disjoint or pairwise disjoint , if any two events are mutually disjoint .
Liang Zhang (UofU)
June 10, 2008 9 / 10
Sample Spaces and Events
Venn Diagrams: e.g.
A ∪ B A ∩ B
A complement mutually disjoint
Liang Zhang (UofU)
June 10, 2008 10 / 10
