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# Probability

150 150 Deborah

### Overview

Usually, probability is calculated in terms of relative numerical frequencies. However, it can also be calculated in terms of ratios of area.

### Probability of Area

Suppose that a target is located on a board. A dart thrown at the board will land somewhere on the board, either outside the target or inside the target. The relative probability that the dart will land somewhere in the area of the target can be calculated by comparing the percentage of the board that is covered by the target with the total area of the board. For example, if the area of the entire target board were 400 square feet and the area of the target was 250 square feet, the probability that a dart would land somewhere in the target would be 250/400, or 0.625.

Figure 1: The probability that the dart will land somewhere in the target is the measurement of the total area minus the area of the target.

### Probability of Length

The ratio of lengths can also be compared to calculate probability. Suppose that a stretch of highway is 100 miles long, and accidents occur randomly along this route. It is not possible to predict that an accident will occur at a precise point along the 100-mile stretch of road, but it is possible to predict the ratio of accidents along an area of highway. The probability that an accident will occur within a 6-mile area of the highway would be 6/100 or .06. The probability that an accident would occur within a 30-mile stretch will be .30.

### Probability of Angle

A fair spinner is constructed so that the spinner is equally likely to point in any direction along the radius of the circle. The circle is divided into wedges from the center of the circle, so that it resembles a pie chart. The probability that it will fall within any wedge will be equal to the ratio of the size of the angle to the entire 360o circle. Suppose that a spinner is constructed so that region A is a 30o angle, region B is a 60o angle, region C is a 90o angle, region D is a 64o angle, and region E is an 116o angle. The probability that the spinner lands in Region B is 60/360, or about 0.17.

Figure 2: The probability that a spinner will land in Region A is about .08, Region B, about 0.17, Region C, 0.25, Region D, about 0.18, Region E, about 0.32

### General Rule

The general rule for calculating probability without counting in a region is by calculating the ratio of the target region to the entire area, if all points occur randomly in an area. There are many applications of this idea. Suppose that the signals from a lost plane are coming from a certain region, but its exact location is unknown. It is possible to predict the possibility that it is within a designated area by comparing the ratio of the search area with the entire area. Until the lost plane has been found, nothing is certain.

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#### Math Introduction to Probability

150 150 Deborah

Overview:

The probability of an event is determined by how likely that event is to happen.  If an event is impossible, it has a probability of 0.  If an event is certain to happen, it has a probability of 1.  Probabilities are usually expressed as fractions, as decimals, or as percentages.

What Is an Event?

An event is a set of outcomes.  There are only two possible outcomes if a coin is tossed.  It can either come up heads or come up tails.  In contrast, if only one die is tossed once, there are six possible outcomes — either a 1,2,3,4,5,or 6.  If one card is chosen out of a deck of standard cards, there are 52 possible outcomes.

What Is a Random Outcome?

An outcome is considered random if every outcome in a set is equally likely.  It is very important that specific instances happen randomly, and that the die, coin, or card is fair or unbiased.  Experimenters go to a great deal of trouble to make sure that subjects within an experiment are assigned to conditions randomly, so that any possible conditions are equally likely to occur.  For the purposes of this introduction, it means that it would be equally possible to get any value within the range of possible outcomes.

What Is the Probability of an Outcome?

The probability of an event is the ratio of successful outcomes to possible outcomes.  For example, when one die is tossed once, there if a probability of 1/6 that the toss will be a 4.  Only one outcome is successful out of 6 possibilities. Suppose the successful outcome is that the toss will result in an even number, a 2, 4, or 6.  Then 3 outcomes are successful out of 6 probabilities, or there is a probability of 3/6.

How Does a Second Outcome Affect Probability?

If both outcomes are independent of each other, that means that one instance has no effect on the other.  (In technical terms, that is known as “sampling with replacement”. ) Given a fair pair of dice, if one rolls the first die and gets a 6, the probability is 1/6.  What is the probability that the second roll will be a 6 also?  For each of the 6 options, there are 6 more options, so the probability would be 1/6 ∙1/6 = 1/36.

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#### Biology Review of Genetics and Probability

150 150 Deborah

Overview:  Chance and Probability
Geneticists use the principles of chance  and probability to express the results of genetic experiments with pea plants, as well as other plants and animals.  In mathematical terms, probability refers to the fraction (or percentage, in decimals) of times an event can occur.  According to the product rule for independent events. the probability of events occurring together is the product of the probabilities for each event multiplied.  For example, the probability of a girl being born is 50 % , but the probability for two girls being born in the same family is 50% x 50% =25%.  This does not necessarily mean that a specific family will have a girl if another family has a boy at the same time, just that the overall probability, in a sample of many, will be 50%.

Predicting Genetic Crosses
In Mendel’s experiments, seven simple genetic factors were manipulated, such as whether the seed is round or wrinkled, the pod is green or yellow, and so on.  At the time those experiments were formed, the way that alleles separate during meiosis and other details of cellular reproduction were not known.  Geneticists assume that alleles segregate during meiosis in a 1:1 ratio, and that gametes combine randomly.  Suppose the dominant gene for round seeds is called R and the recessive form ( for wrinkled seeds) is called r.  One heterozygous parent with phenotype Rr mated with another heterozygous parent with phenotype Rr will have offspring with the probability (1/2 R +1/2 r)(1/2 R + 1/2r) or 1 RR, 2 Rr, 1rr for genotypes, and 3 with round seeds and 1 with wrinkled seeds.  Geneticists often use thousands of trials to produce as many offspring as they can to use the rules of probability,

Punnett Squares and Testcrosses
The range of probable genetic crosses can be demonstrated by drawing a simple grid with all the alleles contributed by one parent on one side and the other parent on the other, then showing each possibility in each square of the grid.  That way, all the possibilities can be accounted for.  http://schooltutoring.com/wp-content/uploads/sites/2/2013/07/punnett-square-green-peas.jpg

Both parents, in this case, contribute G and g.  In addition, biologists can perform a testcross if they have the phenotype to discover the genotype.  In the example above of the green pods, the genotype could be GG or Gg.  By crossing it with a plant with yellow pods (gg), if the genotype is GG, all the offspring will have genotype Gg and have green pods.  If the genotype is Gg, half the offspring will have genotype Gg, with green pods.  The other half will have genotype gg with yellow pods.

Monohybrids and Dihybrids
Some hybrids have only one trait, involving one pair of alleles, and others involve separate pairs of alleles and two separate traits.  When only one trait is involves, the possibilities are simple, and the Punnett square is a simple 2 X 2 grid.  A relatively simple trait in humans is whether earlobes are free (dominant) or attached (recessive).  If two individuals are heterozygous for the trait, their offspring might have the following genotypes:  EE, Ee, Ee, and ee.  If two separate traits are involved, the Punnett square will be larger because of the many different possibilities.  For watermelons, solid color green (G) is dominant over striped green (g), and short length (S) is dominant over long (s).  If a green , short watermelon (GGSS) is crossed with a striped, long watermelon (ggss), the grid grows to 4 X 4, with 16 different genotypes in the second generation.

Incomplete Dominance
Not all alleles are dominant or recessive, however.  Sometimes, both alleles are expressed.  For example, short tailed cats have one gene for no tail(TN) and one gene for long tail (TL).  If two short tailed cats are crossed, the Punnett square has a simple 2 X 2 grid, and the offspring probabilities include 25% long tail (TLTL), 50% short tail (TNTL), and 25% no tail (TNTN).  The roan (reddish-brown) color of some cattle also results from incomplete dominance, as well as many other features of plants and animals.

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#### Who’s Afraid of the Normal Curve?

150 150 Deborah

Overview:  What Is the Normal Curve?
The normal curve is a frequency distribution with special statistical properties.  The best-known application of the normal curve is the distribution of intelligence as measured by tests such as the Stanford-Binet, but there are other examples  that are close to the normal curve in human and animal behavior.  Biologists, other scientists, and mathematicians try to make it scary by calling it a “Gaussian distribution”. To social scientists and educators, it’s nothing but the normal curve.

What Does the Normal Curve Look Like?
The normal curve is a symmetrical distribution of scores with the mean (the average of all scores), the median (the point at which exactly half of all scores are below and the other half of scores are above), and the mode (the most frequent score) are equal.  The further scores get away from the mean, the closer they get to the baseline, but they never touch, even if the score is infinitely far away from the mean.  That is a property of the normal curve called “asymptotic.”  It is also a continuous curve.

Normal Is a Statistical Artifact
The normal curve itself is a theoretical distribution that first came from mathematicians interested in probability and observational errors, and Gauss was one of those mathematicians who developed it.  Scores in real life, such as measurements on IQ tests, ability scores, and the like, follow the normal curve closely, but do not fall exactly on the normal curve.   The mode is the most frequently occurring score in the distribution by definition, and a statistical artifact is a fancy way of saying that the scores follow a certain pattern.

The relative frequencies of scores that are in an approximate normal distribution fall under specific percentages, and no matter what is being measured, if those values follow a normal distribution, those percentages will hold true.  Statisticians call the measurement of the distance away from the mean the standard deviation, and about 68% of scores will fall  within one standard deviation on either side of the mean.  Just a little over 95 % will fall within two standard deviations on either side of the mean, and just over 99% will fall within three standard deviation on either side of the mean.  Because the tails never touch the X-axis, that’s as close as it gets.

Using the Normal Curve in Research
The mathematical properties of the normal curve make statistics easier to apply in research when the values that are obtained from observations fall close to the normal curve.  There are extensive statistical tables that describe the exact location of values on the normal curve.  In addition, because the normal curve was originally developed around probabilities, it is possible to determine how likely something will happen, such as winning the lottery.

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#### Probability Distribution: Random Variables

A random variable is a function which is usually denoted by X defined on the sample space S whose range is the set of real numbers.

i.e. X:S–> R is called a random variable.

Example:

If an unbiased coin is tossed,

Then sample space S={H, T}

Let X denoted the number of tails obtained.

Then X(T)=1 (the number of chances of getting ‘T’ =1).

Probability  distribution:

The probability distribution of a random variable x is as follows.

 X = r 1 2 3 ……. n P(X=r) P(X=1) P(x=2) P(X=3) ………….. P(X=n)

Here all probabilities are non – zero and sum of all probabilities is 1.

Example – 1:

If 2 coins are tossed and X is the random variable which denotes the number of heads then

Sample space, S = {HH,HT,TH,TT}

n(S) = 4.

X can be 0,1,2

P(X=0) = probability of getting zero heads =  1/4 (TT)

P(X=1) = probability of getting 1 head =  2/4=1/2  (HT,TH)

P(X=2) = probability of getting 2 heads =  1/4   (HH)

So, the probability distribution of X is as follows.

 X = r 0 1 2 P(X=r) 1/4 1/2 1/4

Example-2:

If  3 coins are tossed and X is the random variable which denotes the number of heads then

Sample space, S = {HHH,HHT,HTT,TTT,TTH,THH,HTH,THT}

n(S) = 8.

X can be o,1,2,3

P(X=0) = probability of getting zero heads =  1/8 (TTT)

P(X=1) = probability of getting 1 head =  3/8  (HTT,THT,TTH)

P(X=2) = probability of getting 2 heads =  3/8  (HHT,HTH,THH)

P(X=3) = probability of getting 3 heads =  1/8  (HHH)

So, the probability distribution of X is as follows.

 X = r 0 1 2 3 P(X=r) 1/8 3/8 3/8 1/8

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#### Multiplication Theorem on Probability

The probability of happening an event can easily be found using the definition of probability. But just the definition cannot be used to find the probability of happening of both the given events. A theorem known as “Multiplication theorem” solves these types of problems. The statement and proof of “Multiplication theorem” and its usage in various cases is as follows.

Multiplication theorem on probability:

If A and B are any two events  of a sample space such that P(A) ≠0 and P(B)≠0, then

P(A∩B) = P(A) * P(B|A) = P(B) *P(A|B).

Example:  If P(A) =  1/5  P(B|A) =  1/3  then what is P(A∩B)?

Solution: P(A∩B) = P(A) * P(B|A) = 1/5 * 1/3 = 1/15

##### Independent events:

Two events A and B are said to be independent if there is no change in the happening of an event with the happening of the other event.

i.e. Two events A and B are said to be independent if

P(A|B) = P(A) where P(B)≠0.

P(B|A) = P(B) where P(A)≠0.

i.e. Two events A and B are said to be independent if

P(A∩B) = P(A) * P(B).

Example:

While laying the pack of cards, let A be the event of drawing a diamond and B be the event of drawing an ace.

Then P(A) =  13/52 = 1/4 and P(B) =  4/52=1/13

Now, A∩B = drawing a king card from hearts.

Then P(A∩B) =  1/52

Now, P(A/B) = P(A∩B)/P(B) = (1/52)/(1/13) = 1/4 = P(A).

So, A and B are independent.

[Here, P(A∩B) = =    = P(A) * P(B)]

Note:

(1)    If 3 events A,B and C are independent the

P(A∩B∩C) = P(A)*P(B)*P(C).

(2)    If A and B are any two events, then P(AUB) = 1-P(A’)P(B’).

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The probability of happening an event can easily be found using the definition of probability. But just the definition cannot be used to find the probability of happening at least one of the given events. A theorem known as “Addition theorem” solves these types of problems. The statement and proof of “Addition theorem” and its usage in various cases is as follows.

Mutually exclusive events:

Two or more events are said to be mutually exclusive if they don’t have any element in common. i.e. if, the occurrence of one of the events prevents the occurrence of the others then those events are said to be mutually exclusive.

Example:

The event of getting 2 heads, A and the event of getting 2 tails, B when two coins are tossed are mutually exclusive.

Because A = {HH}; B = {TT}.

Mutually exhaustive events:

Two events are said to be mutually exhaustive if there is a certainty of occurring at least one of those two events. i.e. one of those events will definitely happen.

If A and B are two mutually exhaustive then the probability of their union is 1.

i.e. P(AUB)=1.

Example:

The event of getting a head and the event of getting a tail when a coin is tossed are mutually exhaustive.

If A and B are any two events then the probability of happening of at least one of the events is defined as P(AUB) = P(A) + P(B)- P(A∩B).

Proof:

Since events are nothing but sets,

From set theory, we have

n(AUB) = n(A) + n(B)- n(A∩B).

Dividing the above equation by n(S), (where S is the sample space)

n(AUB)/ n(S) = n(A)/ n(S) + n(B)/ n(S)- n(A∩B)/ n(S)

Then by the definition of probability,

P(AUB) = P(A) + P(B)- P(A∩B).

Example:

If the probability of solving a problem by two students George and James are 1/2 and 1/3 respectively then what is the probability of the problem to be solved.

Solution:

Let A and B be the probabilities of solving the problem by George and James respectively.

Then P(A)=1/2 and P(B)=1/3.

The problem will be solved if it is solved at least by one of them also.

So, we need to find P(AUB).

By addition theorem on probability, we have

P(AUB) = P(A) + P(B)- P(A∩B).

P(AUB) = 1/2 +.1/3 – 1/2 * 1/3 = 1/2 +1/3-1/6 = (3+2-1)/6 = 4/6 = 2/3

Note:

If A and B are any two mutually exclusive events then P(A∩B)=0.

Then P(AUB) = P(A)+P(B).

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#### Cognative Friday: Be Reasonable!

150 150 Suzanne

How many times have we all heard the phrase, “Be reasonable!”?

This is, of course, one of the goals for teaching children and so, thinking about how we teach reason, or the act of being reasonable, will go a long way in accomplishing this oft-wished for outcome.  Reasoning is derived through the interplay of different areas of the brain. First, we must process the visual, auditory or other sensory cues which produce the information we must be reasonable about. Next, this information will process through memory and emotional centres of the brain. This is likely the source of our inability to “be reasonable.”

Our memories affect our ability to learn. Positive memories can influence successful learning and negative memories can have the opposite effect. Strong, stark, funny and emotional memories are useful in enhancing learning, but such techniques must be well thought out to be effective. The goal is to avoid associating negative experiences or memories with the learning process. In the event that a child feels negative about a subject, understanding the source of that feeling, and strategizing ways to overcome it can be quite beneficial.

Our emotional centre is activated by the process of memory formation and recall, and so attempts to elicit emotion in learning situations can have positive or negative outcomes. Teaching students a lesson that imparts a profound emotion – one that ranges toward a universal understanding – can be highly effective; however with students’ diverse backgrounds care must be given to the choice of topics used to access this. For instance, a unit on war or urban violence can raise memories of a violence that a student is aware of and may have been effected by. The handling of teachable moments involving difficult issues should certainly not be avoided, but proper introduction and vetting of an issue is essential if it is to be used in a classroom.

And so, once memory and emotion is well-considered, how do we teach reason? The first method is in the asking and complete answering of questions. The important word here is “because.” If a student expresses a thought, make sure that each and every time it is followed by their “because” rationale. This allows for the expansion of the discussion and in-depth analysis can begin. This brings about the process of reasoning – looking at the pros and cons as you and others see it.

Another strategy for teaching reason is having students explain a thought that they do not agree with. This really expands the potential for reasonable discourse, because the student must get into the shoes of “another” and explain why they may hold their beliefs. Understanding through reason is achieved through such an exercise. It may also help a student give more time to understanding the reasoning of others.

Finally, challenge your children and students. If they state an idea, belief or wish, maintain the conversation and ask questions about the possibilities of what they could do with their ideas. Returning to the idea at different times and within different contexts can bring out broader awareness of the reasons for the idea or belief and offers an excellent basis for reflection.

Reason is important to students. They want to understand why they must learn things, what purpose it will serve them and then will ask scores of other questions in a day. Providing answers with reasons is always the first step in teaching the very practical art of “being reasonable.”

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#### Cognitive Friday: Maximizing Learning Styles

150 150 Suzanne

A learning style inventory is an assessment that shows people the modes in which they learn best. It is not an assessment in the common high-stakes sense of the word, but instead a tool to provide information to teachers and learners.  Students learn using one of three dominants modes – their visual brain, their auditory brain or their kinesthetic brain; the Learning Style Inventory can help you recognize which is your style, and which is not. The latter point may be particularly important for some students who do not do well learning in the way that their teacher wants them to learn.

Imagine a teacher in the front of the room reading from a list of 50 common words – Cold, Juice, Level, Hard, Cat and so on. As the teacher reads, students check a box in one of four columns:
1) I see the word,
2) I see an image of the word,
3) I know the sound of the word,
4) I have a physical feeling when I hear the word.  For each word, the child marks which of these four events occur for them. The first two are most common for visual learners, the third is a technique used by auditory learners, and the fourth is the natural response of someone who is kinesthetic.

Many teachers take the time at the beginning of the year to use this assessment with their whole class as a way to plan instructional approaches. If a third of the class is kinesthetic learners, the teacher knows that hands-on learning experiences will be important to incorporate into teaching.  However, the dominant learning style is visual learning, and classrooms are generally set up for that modality. This can cause real problems for students who are not primarily visual learners.

This information helps to understand what to do and what not to do while teaching children. For instance, a kinesthetic learner may have difficulties learning in an environment too full of visual stimuli. Endless posters in a classroom may provoke a range of feelings for a student, which could be highly distracting. A kinesthetic learner needs to know how to channel distractions and maximize the visual and auditory clues they receive. Likewise, a visual learner may be uncomfortable with hands-on learning and may shy away from those learning opportunities.

Even if a teacher does not do a formal learning style inventory with their students, there are ways that a parent can discern this through simple conversation. Just asking what happens in their child’s mind when they hear the word “cat” can start a conversation in which learning style – and learning obstacles – can become much better understood. Taking the time to understand how a child responds to stimuli and cues will go a long way in developing stronger study skills among all students.

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#### Priming and Decision-Making

150 150 Suzanne

The psychological phenomenon of priming affects a number of mental processes. Priming is like an advanced notification system, however a person is receiving notifications that affect what they think and how they feel. Priming is done through the words we hear and see, and the messages we are surrounded by. It is ubiquitous and natural and students can benefit from the messages they are primed with.

Priming can impact decision-making, and so at the points in which students are asked to make decisions, the messages they receive are critically important. When students choose electives or extracurricular activities, the messages they receive about trying something different may occasion a learning opportunity otherwise missed. When thinking about colleges, the many message students receives provide a positive influence on their college-oriented thinking.

Another way that this phenomenon can work is through creating experiences for students which will mirror future experiences that might anticipate. Such role-playing opportunities allow students a chance to make good and bad decisions and reflect on them. They will be better prepared for situations in the future insofar as understanding the implications of the decisions that they make.

One potential pitfall with this aspect of unconscious messages is the ways in which social issues and current events are treated. As students become more aware of the world they live in, the tone taken in an examination of a current event can help to shape the way in which a student will see the world. Political and social values will be created in classroom and learning experiences, and the way these ideas are framed will have an impact.

As students work their way through the transitions that school calls for – moving from elementary to middle to high school – offers many opportunities for students to make decisions that will impact their future. At each point, it is important to remember that the people and environment guiding those decisions will have a strong impact.

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