Probability and Statistics

Linear regression

find the gradient and the y intersect, y = mx + c Use sample points, then simultaneous equations to find a and b

Example 1 - Normal Distribution

if given the percentile and 2 intervals we can calculate: 1. mean 2. Standard Deviation

Bayes' Theorem

is a way of finding a probability when we know certain other probabilities. Like using google search. First think "AB AB AB" then remember to group it like: "AB = A BA / B"

Segmenting a normal distribution curve

it is most commonly segmented in the following ways. 1. Percentile - 100 equal parts 2. Standard Nine - nine parts 3. Quartile - 4 equal parts

key words - Principle of Counting

key words: 1. How many ways, number of ways, total outcome, total number 2. combination

Standard nine (stanine)

score is a way to scale scores on a nine-point scale. It can be used to convert any test score to a single-digit score.

standard deviations from the mean

similar term: Standard Score, Sigma, Z - Score

Permutation word association

1. (specific) Names, Persons, Codes 2. no repetition, only once, Mostly no replacement, 3. order matters - who is selected first

Binomial Distribution vs Mutually exclusive events

1. Mutually exclusive - requires the person to make a choice in the event 2. Binomial is just looking at the outcome of the event. the possibility of getting a 2

Combination (Combination + Arrangement)

1. Out of 7 consonants and 4 vowels, how many 'ways can you select' 3 consonants and 2 vowels? 7C3 x 4C2 = Ans *the order of selection doesn't matter 2. Out of 7 consonants and 4 vowels, how many 'words' of 3 consonants and 2 vowels can be formed? * After the letters are selected, how many ways can the word be arranged. 7C3 x 4C2 x 5! = Ans

Normal Distribution Curve

1. The data is around the mean/mode/median

Arrangement only question

1. There is no choice for a smaller group. 2. when you want items to appear beside each other, group them as one. Que. 1. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together? 2. How many words can the word 'OPTICAL' form.

Word association - Combination

1. These incorporate arrangements (factorial !) where a selection is made from a larger group 2. Order of the group is not specific - committee, team, when number repetition is allowed - Lock code

Mutually Exclusive events (Venn Diagram)

1. analyze equation to see if it has 'AND', 'OR' 2. some questions may require drawing the venn diagram and solving for the unknown. P(A and B) = P(A) + P(B) - P(AnB) Question There are 30 children in a class and they all have at least one cat or dog. 14 children have a cat, 19 children have a dog. What is the probability that a child chosen at random from the class has both a cat and a dog?

Combination (that includes 2 different groups, 1 arrangement)

1. find the combination for each group and 'x' or '+' them. 2. if this selection also forms a word. An arrangement must be applied then the counting principle. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

Standard Deviation and Variance

1. find the mean 2. take all these x values away from the mean and square the results 3. find the square root of this usually the variance is needed first. these will always be positive numbers

Traits of Normal Distribution

1. for the bell curve, there is a continuous random variable. 2. The 1st, 2nd and 3rd Standard Deviations are- 68%, 95% and 99.7%. *remember it is 68% from -1 to 1 3. 'Normally Distributed' means it shapes like a bell.

Binomial Distribution - Tricks

1. if it want the probability of 2 head, then only find (P = 2) 2. if they want the probability of AT LEAST 2 heads (P = 0) x (P = 1) x (P = 2)

Percentile and graphs P. 2

1. look at where the curve starts to measure the percentile growth. Then add where there was no growth. 2. these 2 figures will give the true percentile Tip: only the curve represents percentile decline The line graph shows how the population of parrots on an island declined over the ten year period from 2001 to 2010. Measurements were taken at the beginning of each year. For the period shown in the graph, estimate the time when the population reached its seventy percentile.

Bernoulli Trial

An experiment whose outcome is random and considered: A success 'P' or Failure '1 - p' or 'q'

Percentiles and graphs

1. look at where the curve starts to measure the percentile growth. Then add where there was no growth. 2. these 2 figures will give the true percentile Tip: only the curve represents percentile growth Eg. The population of a town was recorded every twenty years from 1900 to 2000. The results are shown in the line graph. For the period shown in the graph, estimate the year when the population reached its eighty percentile.

Quartile

1. organise the data chronologically. If given frequency, all the numbers must be listed. 2. Dividing the sample set into 4 parts by first using the median (middle value) to find Q2. Then from the lowest value to Q2 to find Q1 Quartile 1 (Q1) can be called the 25th percentile Quartile 2 (Q2) can be called the 50th percentile Quartile 3 (Q3) can be called the 75th percentile

Combinations (at least)

1. this will require looking at the various selection and adding them. At least 1 boy. (2 boys x 1 girl) + (1 boys x 2 girls).... Eg. In a group of 6 boys and 4 girls, four children are to be selected such that at least one boy should be there. *it is general because specific names arn't given

Example of a Dependent Event

2 blue and 3 red marbles are in a bag. What are the chances of getting a blue marble on the second attempt, if a blue marble was selected first?

factorial notation

5! = 5 x 4 x 3 x 2 x1 0! = 1 Arrangement, ways - key word An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf? 8!

Understanding Cards

52: possible outcome 26: black and red cards each Black (13) : Club and Spade Red (13) : Heart and Diamond 4 groups with 13 cards each Clubs, Spades, Hearts, Diamonds

Conditional Probability example 2

70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?

Permutation Trick

A box contains 100 items of which 4 are defective. what is the probability of selecting 1 defective and 1 non-defective if the first item is not replaced? Tip: the ORDER gives 2 choices. If the word 'then' was used it would be 1 choice. *Permutation because all the defects are different Ans: P(1 defective) + P(1 non defective) OR P(1 non defective) + P(1 defective) The total arrangement of the 100 items is not needed

Binomial Distribution - Example

A fair die is thrown four times. Calculate the probabilities of getting: - 2 first and 2 second, therefore it takes into account order 0 Twos 1 Two 2 Twos 3 Twos 4 Twos

Permutation (with replacement)

A password consists of two letters of the alphabet followed by three digits chosen from 0 to 9. Repeats are allowed. How many different possible passwords are there? *Permutation because (A, B, 1, 1, 1) is different from (B, A, 1, 1, 1) The number of ways of choosing the letters = 26 × 26 = 676 The number of ways of choosing the digits = 10 × 10 × 10 = 1,000 So the number of possible passwords = 676 × 1,000 = 676,000

Fundamental Counting Principle- Question

A restaurant offers 5 choices of appetizer, 10 choices of main meal and 4 choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming all choices are available, how many different possible meals does the restaurant offer?

Combination

A selection of objects in which order is NOT important is called a combination. (1, 2, 3) = (3, 2, 1) Each selection is treated like an Independent Event. *it is not as specific as Permutation The number of combinations of n objects taken r at a time is written C(n, r). (n - r) - don't matter r! - the order of r doesn't matter Common Terms - randomly,

Percentile - group data

Add up all percentages below the score, plus half the percentage at the score.

Understanding different Events

All these are based on OUTCOMES Mutually exclusive - when 2 events cannot occur at the same time, dice and cards Inclusive events - when both things can occur at the same time, drawing a king and spade TREE diagram can be used for these Independent event - when the selection of one does not affect the section of the other, item is replaced Dependent event - where the section of one affects the possibility of selecting another, item in NOT replaced. Look at what was selected first Conditional Probability - one event has occurred after a given event, that is A occurs after B. By itself this question requires the formula.

Arrange vs Outcomes

Arrange - this is how many 'WAYS' something can be arranged. think '!' (notation) arrangement of 6 numbers without the importance of order is, 6! = 720 Outcomes - this is the result of a trial. which can also be the sample space. (Counting principle) Total outcome of 2 dice = 6 x 6 = 36

Permutation and letters 2

Assuming that any arrangement of letters forms a 'word', how many 'words' of any length can be formed from the letters of the word SQUARE? *any length = 1 letter word, 2 letter word etc *letters chosen cannot be replaced

Independent Events - Example

At a picnic, Julio reaches into an ice-filled cooler containing 8 regular soft drinks and 5 diet soft drinks. He removes a can, then decides he is not really thirsty, and puts it back(this is assumption with independent questions). What is the probability that Julio and the next person to reach into the cooler both randomly select a regular soft drink?

Terms part 2

Combination this is a special arrangement A - if the selection is from a larger group, look whether order is important or not. Then select Permutation of Combination B - if there is not larger group use counting principle Arrangement is more general - ! Eg. how may ways can you arrange 5 books

Permutation VS Combination

Combination: A group of seven students working on a project needs to choose two students to present the group's report. How many ways can they choose the two students? Permutations: A group of seven students working on a project needs to choose two students to present the group's report. How many ways can they choose the Julian and Kemoy? Permutations are for lists (order matters) and combinations are for groups (order doesn't matter).

Word Association P2

Counting principle - how many ways, total outcome, n(AUB) = n(A) + n (B) - n(AnB) Counting principle - outcome when 2 things are together, it is considered 1

When to use: mean, median or mode

Example: SWEEPSTAKES A sweepstakes offers a first prize of $10,000, two second prizes of $100, and one hundred third prizes of $10. Which measure of central tendency best represents the available prizes? Since 100 of the 103 prizes are $10, the mode ($10) best represents the available

Permutation and letters

How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed? *no repetition = Permutation

Arrangement with repetition

How many different ways can the letters of the word MISSISSIPPI be arranged? The letter I occurs 4 times, S occurs 4 times, and P occurs twice. *since the S, I, P can be arranged in any order (for example it doesn't matter which P you put first) - take them from the equation

Relative Frequency

How often something happens divided by all outcomes. Total sum is 1 (it is also connected to the probability of something happen) Your team has won 9 games from a total of 12 games played. the Frequency of winning is 9. the Relative Frequency of winning is 9/12 = 75%

Permutation - At least

If asked for 'at least' 2 out of 20. 1. find 2 up to 20 2. find the total arrangement - 0, 1 and 2.

Dependent Events

If the outcome of an event does affect the outcome of another event, the two events are dependent. Look for the subordinate conjunctions: before, after, given or also ( in this case look like what happens first) TREE diagram/ Formula can be used for this Taking a piece of candy from a jar and then taking a second piece WITHOUT replacing the first are dependent events because taking the first piece affects what is available to be taken next.

Arrangement with grouped words

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together? 1. CRPRTN(OOAIO) 2. (OOAIO) is considered 1 letter 3. since there are 3 'Os' the order doesn't matter

Central Limit Theorem

In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.

sum of squares

It is defined as being the sum, over all observations, of the squared differences of each observation from the overall mean.

Mutually Exclusive example

It is known that the probability of obtaining zero defectives in a sample of 40 items is 0.34, whilst the probability of obtaining 1 defective item in the sample is 0.46. What is the probability of (a) obtaining not more than 1 defective item in a sample? 0.34 + 0.46 = 0.8 (b) obtaining more than 1 defective items in a sample? 1 - 0.8 =

Permutation with condition

Jones is the Chairman of a committee. In how many ways can a committee of 5 be chosen from 10 people given that Jones must be one of them? Jones is already chosen, so we need to choose another 4 from 9. Therefore, P(9, 4)

Mean Deviation

Mean Deviation tells us how far, on average, all values are from the middle

Probability with Combinations

Monifa has a collection of 32 CDs—18 R&B and 14 rap. As she is leaving for a trip, she randomly chooses 6 CDs to take with her. What is the probability that she selects 3 R&B and 3 rap?

Norm distribution vs Standard Norm distribution

Norm distribution - the mean is used at the centre Standard norm distribution - 0 is at the center

Venn diagram and probability

P (A n B) = A and B = A x B P (A u B) = A or B = A + B

Notation of Conditional Probability

P(B|A) means - the probability of Event B occurring given that Event A has already occurred. In most examples, this means that an item is replaced

Permutation - Trick 3

Permutation usually means something can be used 'only once' eg. The number of ways of choosing 4 letters from 26.

Probability

Possible options / total Options. Key words: possibility, probability,

Event with 2 outcomes

Probability of tossing a coin 3 times. 1. look at the outcomes: only 2 2. look at how many times the action is done

Terms in Combination and permutation

Sample size - total number or outcomes.

Permutation question

Suppose a coach has 7 possible players in mind for the top 4 spots in the lineup. *Tip: player = permutation, team = combination simple answer: there are 7 ・ 6 ・ 5 ・ 4 or 840 ways that she could assign players to the top 4 spots. Proper answer: from the 7 you can only select 4, the other 3 you can't select you must take away those options by dividing

Standard Deviation

The Standard Deviation is a measure of how spread out numbers are. it gives us a RANGE within the mean to operate

Percentile

The data is divided into 100 equal parts. a person who score falls in the 65th percentile, has outperformed 65% of those who have taken the test. The percent of cases that are at or below a score. Always put the data in chronological order. This is like the cumulative score

Dependent Events - Diagram

The notation reads, 'A has occurred by itself x B has occurred AFTER A'. For example, after a student who does Math (A) he then selects Spanish (B). TREE diagram/ Formula can be used for this Both events must occur in a manner where one was selected first then the other, thus synonymous with NOT replacing. P(A and B) = P(A) ・ P(B following A).

Interpreting percentile graphs

The percentile always represents the OPPOSITE of what the graph represents • Students' scores increased over 10 years, 65 percentile represent amount of students who did not improve over 65% • 100 students did not increase over 10 years, 65 percentile represent amount of students who did improve over 65%

plotting a deviation curve

The standard deviation σ ≈ 5.594, which is basically 5.6 million people This is the average distribution from the mean, each city has an average population of x ̅ = 5.4, the 1st deviation is ± 5.6 million people. the first deviation will have majority of the population, thus forming a bell

Principle of Counting

The total amount of ways something can be done or combined (there is no selection from a larger group). We have 2 t-shirts and with each t-shirt we could pick 4 pairs of jeans. Altogether there are 2 × 4 = 8 possible combinations. **Combination - the order does not matter (blue jeans and white top) = (white top and blue jeans)

Variance

The variance (σ2) is the average of the squared differences from the Mean. Here is how it is defined: also called dispersion. The simplest variance is the Range. Associated words: Range, standard deviation if it is an entire population, it is 'N', if it is a sample size, it is 'n - 1'

Arranging items in a circle

There are (n−1)! ways to arrange n distinct objects in a circle

Regression

This is a form of statistical analysis used to predict the value of y (dependent variable), using the independent variable of x There are different equations, which can be derived dependent of the shape of the graph - straight line, logarithm, exponential

Correlation

This is looking at the data between 2 things Correlation is Positive when the values increase together. Love and happiness. (the data is proportional) Correlation is Negative when one value decreases as the other increases. Wealth and poverty (the data is not proportional) These correlation work best with straight lines Values range from: -1, 0, 1

Percentile - when given small data only

This is usually simple like 25%, 50% 1. Put in chronological order 2. find the place of the required figure 3. place this over the total figures Tip: if given a frequency diagram, list all the numbers chronologically

Converting a Normal distribution to a Standard Normal Distribution

This puts the data in a easier manner to interpret. 1. first subtract the mean (mean from the distance) ans will be - or + 2. then divide by the Standard Deviation the answers will give negative values. pay attention to the x-axis

Normal distribution form

Through integration this can be used to form the area under the curve which is 1. this can also be used to determine the probability of the variable. 1. is there a variable x on the normal distribution 2. the z-scores must be calculated 3. is the probability needed

Identifying conditional probability

Tip: 1. look for conjunction 2. Identify what happens first 3. use the formula and identify the 'n' eg. PnB Look for the subordinate conjunctions: before, after, given (in this case look like what happens first) No Coordinating Conjunction 1. Let G denote the event 'student is girl' and let F denote the event 'student is French'. - the probability that if I pick a French student, it will be a girl, that is, P(F|G) Coordinating Conjunction 2. In my town, it's rainy (R) one third of the days. Given that it is rainy, there will be heavy traffic (T). P (T I R)

Percentile - large data

Use a cumulative frequency curve. this is usually simple like 33%, 57%

Permutations (Choice from Bigger Group)

When a GROUP of OBJECTS or people are arranged in a unique ORDER, the arrangement is called a permutation. numerator: Total choices available denominator: order that does not matter Permutation: (1, 2) & (2, 1) - this is considered 2 different things, because 1 is selected first then 2 for the first ordered pairs. Combination: (1, 2) & (2, 1) - the order doesn't matter and this is considered the same thing. n - number of ways something can be arranged r - object/spaces taken at a time

Random

When all outcomes have an equally likely chance of occurring

Inclusive Events

When both things can occur at the same time. probability of drawing a king or a spade from a standard deck of cards

Combination (fixed position)

a) if one position is fixed, like the eldest boy is already selected. Eg. how many ways can you arrange for boys out of 10. then 9C3 b) if he is excluded then he is not a option, 9C4 c) put (a) over number of all possible groups 10C4

Tending to a normal distribution

as the number of samples taken increases, the sample mean becomes closer to the population mean. this is also, central limit theorem

Binomial Distribution

look for scenarios with only 2 outcomes. Eg. coin, chicken, no chicken. 1. The experiment consists of n repeated trials; 2. Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial); 3. The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.

Principle of Counting and Dependent Events

n(AnB) = n(A) x n(B I A) In how many ways can 2 jacks be drawn in succession from an ordinary 52-card deck if the first card is drawn but not replaced. 4 x 3 = 12

Deviation score

once the mean is calculated correctly, the sum of the deviation score is always 0

Conditional Probability example 1

one event has occurred after a given event P (A I B) Let A denote the event 'student is female' and let B denote the event 'student is French'. In a class of 100 students suppose 60 are French, and suppose that 10 of the French students are females. Find the probability that if I pick a French student, it will be a girl, that is, find P(A|B).

Standardize a Normal Distribution question

since the standard deviation is 6.6. the first deviation from the mean is (6.6 - 23) -16.4. then this figure has to be standardised (16.4 - 23/ 6.6) = -1 when the other figures are standardised, anything that falls less than -1 is considered a failure.

Diagram of a normal distribution curve

the mean must be found then the standard deviations. This is a continuous normal distribution curve.

Continuous Probability distribution

the outcome can be any value in a interval of real numbers and an equation is usually needed. the time it takes to come home from work. represented by a curve

Fundamental Counting Principle

the total amount of ways something can be done Words If event M can occur in m ways and is followed by event N that can occur in n ways, then event M followed by event N can occur in m • n ways. Example If event M can occur in 2 ways and event N can occur in 3 ways, then M followed by N can occur in 2 • 3 or 6 ways.

Power regression

these are information done in the form of powers

Logarithmic regression

these shape like a log curve, y = a + b ln x Use sample points, then simultaneous equations to find a and b

Exponential regression

this function models growth or decay b > 1, exponential growth 0 < b < 1, decay Use sample points, then simultaneous equations to find a and b

Z - score

this is (mean from the distance) divided by the standard deviation. 1. 'standardised' the value. When this is done the mean becomes 0 2. use the z table to find the required answer. sometimes we want to find the probability of Z being between certain values.

formula for Correlation

this is not the Gradient. there are many formulas that can be used, here is the most common one

Mutually Exclusive

this is when 2 events CANNOT occur at the same time. Rolling a die and getting a '4' and '5' at once. Therefore you get one OR the other. Sometimes these questions require subtraction from 1. 1. Determine if the events can occur at the same time. 2. Look for words live "no more than" or "at least", "OR" OR = + P(A or B) = P(A) + P(B)

symmetrical outcome

this occurs when the probability of an event happening is 50 / 50 . It forms a normal distribution It will be skewed if this is not the case. like rolling a die that has 6 sides.

Independent Events

when the selection of one does not affect the probability of another section. *to make independent, terms like 'put back', 'take out' are used TREE diagram can be used for this The choices of letters and digits to be put on a license plate are called independent events because each letter or digit chosen does not affect the choices for the others. Many answering machines allow owners to call home and get their messages by entering a 3-digit code. How many codes are possible? 10 x 10 x 10 = 1000

Discrete probability distribution

when there is a finite number of values, sometimes represented by a histogram. eg number of cars in the parking lot

Binomial Distribution - Mean, Variance

μ = np (total number x probability of success) Variance: σ2 = n p(1 - p)