SMHS Probability - Revision history

Glenbrau: /* Problems */

2015-03-24T18:44:21Z

‎Problems

Glenbrau: /* Applications */

2015-03-24T18:32:56Z

‎Applications

Glenbrau: /* Obesity Example */

2015-03-24T18:32:03Z

‎Obesity Example

Glenbrau: /* Obesity Example */

2015-03-24T18:31:37Z

‎Obesity Example

Glenbrau: /* Melanoma Example */

2015-03-24T14:46:05Z

‎Melanoma Example

Glenbrau: /* Contingency Tables */

2015-03-24T14:24:42Z

‎Contingency Tables

Glenbrau: /* Independence vs. disjointness/mutual exclusiveness */

2015-03-24T14:22:29Z

‎Independence vs. disjointness/mutual exclusiveness

Glenbrau: /* Counting */

2015-03-20T19:01:42Z

‎Counting

Glenbrau: /* Event manipulations */

2015-03-20T18:50:37Z

‎Event manipulations

Glenbrau: /* Types of probabilities */

2015-03-20T18:36:56Z

‎Types of probabilities

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 * A coin is tossed 400 times and 170 heads are observed. This coin is __ ?
-: (a) fair, because the probability of seeing that amount of heads or less is approximately 0.0013
+: (a) Fair, because the probability of seeing that amount of heads or less is approximately 0.0013.
-: (b) neither fair nor unfair. There is not enough information to determine that.
+: (b) Neither fair nor unfair. There is not enough information to determine that.
-: (c) fair, because the probability of seeing that amount of heads or less is approximately 0.5
+: (c) Fair, because the probability of seeing that amount of heads or less is approximately 0.5
-: (d) not fair, because the probability of seeing that amount of heads or less is close to 0.
+: (d) Not fair, because the probability of seeing that amount of heads or less is close to 0.
 * If two events are independent, then they are automatically mutually exclusive.

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 * [http://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Prob_Simul This website] introduces an application of probability theory through simulation. Many practical examples require probability computations of complex events. Such calculations may be carried out exactly, using the rules of probability, or approximately using estimation and/or simulations. [http://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Prob_Simul SOCR simulations] may be used to compute approximate probabilities for various processes and to compare these empirical probabilities to their exact counterparts. This article included examples of a ''Ball and Urn Experiment, Binomial Coin Toss Experiment, Card Experiment, Roulette Experiment, and Chuck A Luck Experiment''. It is a valuable source for practicing simulations using probability theory.
-* [http://www.probabilitytheory.info This website] offers a list of interesting articles on the topic of probability theory. It includes a general introduction to the history of probability theory and addresses a wide list of articles of the application of probability in different areas including business, medicine, economics, and biology. These short articles are good starting place to learn about applications of probability theory in various fields.
+* [http://www.probabilitytheory.info This website] offers a list of interesting articles on the topic of probability theory. It includes a general introduction to the history of probability theory and addresses a wide list of articles of the application of probability in different areas including business, medicine, economics, and biology. These short articles are a good starting place to learn about applications of probability theory in various fields.
 ===Software ===

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 </center>
-* Complete the table using the given information
+Complete the table using the given information:
 * Compute the probability that a child has a BMI <27, $ P(Child and BMI<27) $:
 * Compute the probability that a randomly chosen subject is an adult or has a BMI over 30, $ P(Adult or BMI>30) $:

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 * Complete the table using the given information
-* Compute the probability that a child has a BMI <27, P(Child and BMI<27):
+* Compute the probability that a child has a BMI <27, $ P(Child and BMI<27) $:
-* Compute the probability that a randomly chosen subject is an adult or has a BMI over 30, P(Adult or BMI>30):
+* Compute the probability that a randomly chosen subject is an adult or has a BMI over 30, $ P(Adult or BMI>30) $:
-* Compute the probability that a randomly chosen subject is normal (22<BMI<27) given that the subject is an elderly, P(22<BMI<27 | Elderly):
+* Compute the probability that a randomly chosen subject is normal (22<BMI<27) given that the subject is an elderly, $ P(22<BMI<27 | Elderly) $:
 ===Applications===

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 </center>
-* Suppose we select one out of the 400 patients in the study and we want to find the probability that the cancer is on the extremities ''given'' that it is a type of  nodular: P = 73/125 = P(Extremities | Nodular)
+* Suppose we select one out of the 400 patients in the study and we want to find the probability that the cancer is on the extremities ''given'' that it is a type of  nodular: $ P = 73/125 = P(Extremities | Nodular) $
 * What is the probability that for a randomly chosen patient the cancer type is Superficial given that it appears on the Trunk?

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 ====Contingency Tables====
-Contingency tables provide data summaries that can be used to compute various probabilities of interest, e.g., determining conditional or marginal probabilities. These tables display sample values indexed by two different variables that may be dependent (contingent) on one another.
+Contingency tables provide data summaries that can be used to compute various probabilities of interest (e.g., determining conditional or marginal probabilities). These tables display sample values indexed by two different variables that may be dependent (contingent) on one another.
 =====Melanoma Example=====

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 These two concepts are different and should not be conflated. If two events are mutually exclusive, they cannot happen together (i.e.,  $P(A│B)=0)$). The occurrence of one provides information about the probability of the other; therefore, events that are mutually exclusive cannot be independent.
-Consider the [[SOCR_EduMaterials_Activities_PokerExperiment|SOCR poker game]]. If we know the card we picked randomly is a queen, then the event that it is a red queen given that it is a queen and the event that it is a black queen given that it is a queen are independent. The event that it is a black card is not mutually exclusive from the event that it is a spade.
+Consider the [[SOCR_EduMaterials_Activities_PokerExperiment|SOCR poker game]]. If we know the card we picked randomly is a queen, then the event that it is a red queen (given that it is a queen) and the event that it is a black queen (also given that it is a queen) are independent. The event that it is a black card is not mutually exclusive from the event that it is a spade.
 ====Contingency Tables====

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 Counting principles are very useful in probability theory. Consider picking 3 students from a total of 26 students named A to Z.
 *'''Permutation''': Permutation is the rearrangement of objects in distinguishable sequences. Each unique ordering is called a permutation. For example $\{A, B, D\}$ is different from $\{D, A, B\}$. There are $3!=6$ permutations of students A, B and D.
-**Permutation with repetitions (replacement): If the ordering of objects matters and an object can be chosen more than once then the number of permutations is $ n^r $, where n is the number of objects from which you can choose and r is the number of objects you choose. In our example above, we have $ 26^3 $  permutations with repetitions.
+**Permutation with repetitions (replacement): If the ordering of objects matters and an object can be chosen more than once, then the number of permutations is $ n^r $, where n is the number of objects from which you can choose and r is the number of objects you choose. In our example above, we have $ 26^3 $  permutations with repetitions.
 **Permutation without repetitions (replacement): If the order matters and each object can be chosen only once, then the number of permutation is $ n(n-1)…(n-r+1)=\frac{n!}{(n-r)!} $, where $n$ is the number of objects you can choose from and $r$ is the number of objects you choose. In our example above, we have $26*25*24$ permutations without repetitions.
 *'''Combinations''': A combination is an un-ordered collection of unique objects. In our example above, {A, B, D} is the same as {D, B, A}.
-**Combinations with repetitions (replacement): This is the case in which order does not matter, and an object can be chosen more than once. The number of combinations is $ {n+r-1 \choose r}= \frac{(n+r-1)!}{r!(n-1)!} $. In The example above, we have $\frac{(26+3-1)!}{3!(26-1)!}=3276$ combinations with repetitions.
+**Combinations with repetitions (replacement): This is the case in which order does not matter, and an object can be chosen more than once. The number of combinations is $ {n+r-1 \choose r}= \frac{(n+r-1)!}{r!(n-1)!} $. In the example above, we have $\frac{(26+3-1)!}{3!(26-1)!}=3276$ combinations with repetitions.
 **Combinations without repetitions (replacement): This is the case in which the order does not matter, and an object can be chosen only once. The number of combinations is $ {n \choose r}=\frac{n!}{r!(n-r)!} $, where $n$ is the number of objects you can choose from and $r$ is the number of objects you choose. In our example, we have $ {26 \choose 3}=2600 $ combinations without repetitions.
 * [[AP_Statistics_Curriculum_2007_Prob_Count| Examples of combinations and permutations applications]].

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 * Intersection: $A\cap B$ contains all outcomes which are in both $A$ and $B$.
 * Mutually exclusive events are events that cannot occur at the same time, $A\cap B =\emptyset$.
-* Conditional Probability: The conditional probability of event $A$ occurring given that event $B$ occurs is $ P(A│B)=(P(A\cap B))/(P(B)) $. If $A$ and $B$ are independent then knowing $B$, or $B^c$, gives no information on the probability of $A$, i.e., $ P(A│B)=P(A) $.
+* Conditional Probability: The conditional probability of event $A$ occurring, given that event $B$ occurs is $ P(A│B)=(P(A\cap B))/(P(B)) $. If $A$ and $B$ are independent, then knowing $B$, or $B^c$, gives no information on the probability of $A$, i.e., $ P(A│B)=P(A) $.
 * Multiplication rule: For any two events, $A$ and $B$, $ P(A\cap B)=P(A│B)P(B) $. In general, for $n$ events $A_1, ..., A_n$: $ P(A_1 \cap A_2 \cap A_3 \cap … \cap A_n ) = P(A_1)P(A_2│A_1)P(A_3│A_1\cap A_2 ) … P(A_n│A_1\cap A_1\cap A_2\cap A_3\cap … \cap A_{n-1} ) $.
 * Law of total probability: $P(B)=P(B│A_1 )P(A_1 )+P(B│A_2 )P(A_2 )+⋯ +P(B│A_n )P(A_n) $, where the events $ {A_1,…,A_n} $ partition the sample space $S$.

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 ====Types of probabilities====
-Probability models have two components: sample space and probabilities.
+Probability models have two components: ''sample space'' and ''probabilities''.
 *The '''sample space (S)''' for a random experiment is the set of all possible outcomes of the experiment.
-**Event: An event is a collection of outcomes.
+**''Event'': A collection of outcomes.
 **An event is said to occur if any outcome making up that event occurs.
-*'''Probabilities''' for each event in the sample space.
+*'''Probabilities''' for each event are included in the sample space.
 **Probabilities may come from models and be mathematical and/or physical descriptions of a sample space and the chance of each event. An example of this would be a fair dice tossing game.
 **Probabilities may be derived from data. Data observations can determine probability distribution. An example would be tossing a coin 50 times and observing the heads count.