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Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random) event will occur.<br> | Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random) event will occur.<br> | ||
| − | + | <br> | |
= '''Discrete Probability'''<br> = | = '''Discrete Probability'''<br> = | ||
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= '''Probability Theory''' = | = '''Probability Theory''' = | ||
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In this section, every outcome might not have the same probability, so assigning probabilities might be necessary. One example of all outcomes not having identical probabilities is in the example of a loaded dice (One number on the die has a larger probability than others). | In this section, every outcome might not have the same probability, so assigning probabilities might be necessary. One example of all outcomes not having identical probabilities is in the example of a loaded dice (One number on the die has a larger probability than others). | ||
| + | <br> | ||
| − | + | <u>'''Assigning Probabilities'''</u> | |
| − | <u>'''Assigning Probabilities'''</u> | + | |
If S is the sample space of an experiment wih a finite number of outcomes, then p(s) is assigned to each outcome s. | If S is the sample space of an experiment wih a finite number of outcomes, then p(s) is assigned to each outcome s. | ||
| − | 2 conditions must be met when assigning probabilities: | + | 2 conditions must be met when assigning probabilities: |
| − | #0<u><</u>p(s)<u><</u>1 for each s that exists in S | + | #0<u><</u>p(s)<u><</u>1 for each s that exists in S |
#the sum of all probabilities of s that exist in S must equal 1 | #the sum of all probabilities of s that exist in S must equal 1 | ||
| + | <br> | ||
| + | '''Example''' | ||
| − | + | <br> | |
| + | <u>Uniform Distribution</u>- Assigns the probability 1/n to each element of S, if S is a set with n elements | ||
| + | <u>The probability of an event (A)</u>- The sum of the probability of all the outcomes of A | ||
| − | < | + | <br> |
| − | <u> | + | <u>'''Probabilities of Complements and Unions of Events'''</u> |
| + | <u>Conditional Probability of A and B p(A | B)-</u> the probability of the union of A and B divided by the probability of B | ||
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| + | p(A | B) = p( A u B)/ p(B)<br> | ||
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| + | <u>Independent</u>- Events A and B are independent if and only if p(A n B)= p(A)p(B) | ||
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| + | '''Example''' | ||
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[Category:MA375Spring2012Walther]<br> | [Category:MA375Spring2012Walther]<br> | ||
Revision as of 11:06, 21 April 2012
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random) event will occur.
Discrete Probability
Discrete probability restricts one to experiments that have finitely many, equally likely, outcomes. There are several terms one must familiarize themselves with before talking about discrete probability, which are listed below:
- Experiment- A procedure that yields one of a given set of possible outcomes
- Sample Space- The set of possible outcomes
- Event-A subset of the sample space
- The complementary event of E- (E)- if E is an event in the sample space S is the subtraction of E from S: E = S-E
- Union of two events A and B (A u B)- The union of two events A and B is the set of outcomes that belong to A or B or both.
- Intersection of two events A and B (A n B)-The intersection of two events A and B is the set of outcomes that belong to both A and B.
Now that there is an understanding of some fundamental definitions, the definition of Probability now can be defined:
The probability of E (if E is an event and S is a finite nonempty sample space of equally likely outcomes), p(E), is equal to |E| / |S|
p(E) = |E| / |S|
Examples:
The probability of the complementary event (E) is given by the following equaiton:
p(E) = 1-P(E)
Example:
The probability of A union B is given by the below formula:
p(A u B)= p(A)+p(B)-P(A n B)
Example
Probability Theory
In this section, every outcome might not have the same probability, so assigning probabilities might be necessary. One example of all outcomes not having identical probabilities is in the example of a loaded dice (One number on the die has a larger probability than others).
Assigning Probabilities
If S is the sample space of an experiment wih a finite number of outcomes, then p(s) is assigned to each outcome s.
2 conditions must be met when assigning probabilities:
- 0<p(s)<1 for each s that exists in S
- the sum of all probabilities of s that exist in S must equal 1
Example
Uniform Distribution- Assigns the probability 1/n to each element of S, if S is a set with n elements
The probability of an event (A)- The sum of the probability of all the outcomes of A
Probabilities of Complements and Unions of Events
Conditional Probability of A and B p(A | B)- the probability of the union of A and B divided by the probability of B
p(A | B) = p( A u B)/ p(B)
Independent- Events A and B are independent if and only if p(A n B)= p(A)p(B)
Example
[Category:MA375Spring2012Walther]
