Revision as of 12:20, 22 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.

Figure 1: A Pictoral Representation of A Union B

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.

Figure 2: A pictoral Representation of A intersect 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| Eq. 1

Example:

Question: A bucket has 4 blue balls and 6 red balls. What is the probability that a ball chosen at random from the bucket will be blue?

Solution: There are 10 possibile outcomes (4 blue + 6 red). 4 of these outcomes will result in a blue ball; so the probability of choosing a blue ball is 4/10 or 40%.

The probability of the complementary event (E) is given by the following equaiton:

p(E) = 1-P(E) Eq. 2

Example:

Question: A sequence of 9 bits is randomly generated. What is the probability that at least one of these bits is a 1?

Answer: E is the event that at least one of the 9 bits is 1. The sample space, S, is the set of all bit strings length 9. Using the

The probability of A union B is given by the below formula:

p(A u B)= p(A)+p(B)-P(A n B) Eq. 3

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) Eq. 4

Independent- Events A and B are independent if and only if the probability of the union of A and B is equal to the probability of A multiplied by the probability of B.

p(A n B)= p(A)p(B) Eq. 5

Example

Bernoullu Trials</u

<u>Bernoulli Trial- Each performance of an experiment where there are only 2 possibilities (usually called a success of failure).

The probability of exactly k successes in n independent Bernoulli Trials- C(n,K)p^k q^n-k , where the probability of success=p and probability of failure= q =1-p

[Category:MA375Spring2012Walther]

@@ Line 11: / Line 11: @@
 *<u>Event</u>-A subset of the sample space<u><br></u>
 *<u>The complementary event of E-</u> (<u>E</u>)- if E is an event in the sample space S is the subtraction of E from S:&nbsp;<u>E</u> = S-E &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <br>
 *<u>Union of two events A and B (A u B)-</u> The union of two events A and B is the set of outcomes that belong to A or B or both.<br>
+[[Image:AunionB.png|center]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;'''&nbsp; Figure 1''': A Pictoral Representation of A Union B<br>
 *<u>Intersection of two events A and B (A n B)</u>-The intersection of two events A and B is the set of outcomes that belong to both A and B.<br>
+[[Image:AintersectB.png|center]]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; '''Figure 2''': A pictoral&nbsp;Representation of A intersect B
 <br>
@@ Line 20: / Line 25: @@
 The <u>probability</u> of E (if E is an event and S is a finite nonempty sample space of equally likely outcomes), p(E),&nbsp; is equal to |E| / |S|<br>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(E) = |E| / |S|<br>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(E) = |E| / |S|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''Eq. 1'''<br>
 &nbsp;
-'''Examples:'''
+'''Example:'''
+''Question'': A bucket has 4 blue balls and 6 red balls.&nbsp; What is the probability that a ball chosen at random from the bucket will be blue?
+''Solution'': There are 10 possibile outcomes (4 blue + 6 red).&nbsp; 4 of these outcomes will result in a blue ball; so the probability of choosing a blue ball is 4/10 or 40%.<br>
-<br>
 The <u>probability of the complementary event</u> (<u>E</u>) is given by the following equaiton:<br>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp;&nbsp;&nbsp; p(<u>E</u>) = 1-P(E)<br>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; p(<u>E</u>) = 1-P(E) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; '''Eq. 2'''<br>
 '''Example:'''
+''Question:'' A sequence of 9 bits is randomly generated.&nbsp; What is the probability that at least one of these bits is a 1?
+''Answer: ''E is the event that at least one of the 9 bits is 1.&nbsp; The sample space, S, is the set of all bit strings length 9. Using the
 <br> The <u>probability of A union B</u> is given by the below formula:<br>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(A u B)= p(A)+p(B)-P(A n B)
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(A u B)= p(A)+p(B)-P(A n B) &nbsp; &nbsp;&nbsp; '''Eq. 3'''
 '''Example'''<br>
@@ Line 71: / Line 84: @@
 <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&nbsp;
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(A |&nbsp;B) = p(&nbsp;A&nbsp;u B)/ p(B)<br>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(A |&nbsp;B) = p(&nbsp;A&nbsp;u B)/ p(B)&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; '''Eq. 4'''<br>
-<u>Independent</u>- Events A and B are independent if and only if p(A n B)= p(A)p(B)
-<br>
-'''Example'''
+<u>Independent</u>- Events A and B are independent if and only if the probability of the union of A and B is equal to&nbsp; the probability of A multiplied by the probability of B.
-<br>
-<u>'''Bernoullu Trials'''</u<br>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; p(A n B)= p(A)p(B)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; '''Eq. 5'''<br>
-<u>Bernoulli Trial</u>- Each performance of an experiment where there are only 2 possibilities (usually called a success of failure).<u><br></u>
+<br>
-<u>The probability of exactly k successes in n independent Bernoulli Trials</u>- C(n,K)p<sup>k</sup> q<sup>n-k</sup>&nbsp; , where the probability of success=p and probability of failure= q =1-p
+'''Example''' <br>
+<u>'''Bernoullu Trials'''&lt;/u<br></u>
+<u>&lt;u&gt;Bernoulli Trial</u>- Each performance of an experiment where there are only 2 possibilities (usually called a success of failure).<u><br></u>
+<u>The probability of exactly k successes in n independent Bernoulli Trials</u>- C(n,K)p<sup>k</sup> q<sup>n-k</sup>&nbsp; , where the probability of success=p and probability of failure= q =1-p
+<br>
+<br>
 [Category:MA375Spring2012Walther]<br>

Difference between revisions of "Mysteries of probability MA375S12Walther" - Rhea

Revision as of 12:20, 22 April 2012

Discrete Probability

Probability Theory

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