Probability Basics

Question 1

What is Independent

Answer 1

each event is not affected by other events

Probability of an event happening =
Number of ways it can happen
/ Total number of outcomes

Question 2

What is Dependent

Answer 2

also called “Conditional”, where an event is affected by other events

A and B = A and (A l B)
“Probability of event A and event B equals
the probability of event A times the probability of event B given event A”

Question 3

What is Mutually Exclusive

Answer 3

events can’t happen at the same time

Question 4

Permutations with Repetition

Answer 4

n^r

where n is the number of things to choose from,
and we choose r of them,
repetition is allowed,
and order matters.

Question 5

Combination

Answer 5

When the order doesn’t matter

Question 6

Permutation

Answer 6

When the order does matter

Question 7

Permutations without Repetition

Answer 7

n!
/ (n − r)!

where n is the number of things to choose from,
and we choose r of them,
no repetitions,
order matters.

Question 8

Combinations without Repetition

Answer 8

n!/(r!(n-r)!)

It is often called "n choose r"
where n is the number of things to choose from,
and we choose r of them,
no repetition,
order doesn't matter.

Question 9

Combinations with Repetition

Answer 9

(r+n-1)!/ (r!(n-1)!)

where n is the number of things to choose from,
and we choose r of them
repetition allowed,
order doesn’t matter.
This is the same as a combination without repetition where n = r + n - 1

Question 10

Standard Deviation definition (not formula!!)

Answer 10

The formula is easy: it is the square root of the Variance. So now you ask, “What is the Variance?”

Question 11

“What is the Variance?”

Answer 11

The average of the squared differences from the Mean.

Var(X) = Σx^2p − μ^2

To calculate the Variance:

square each value and multiply by its probability
sum them up and we get Σx^2p
then subtract the square of the Expected Value μ^2

μ = expected value = Σxp

Question 12

Formula for The “Population Standard Deviation”:
and
The “Sample Standard Deviation”:

Answer 12

Population Standard Deviation
square root of [ (1/N) Σ of (x - mu)^2 ]

Sample Standard Deviation”
square root of [ (1/(N-1)) Σ of (x - x^bar)^2 ]

Weighted mean standard deviation
Square root of [Σ(x^2 (p) − μ^2)]

μ = expected value = Σxp

Question 13

What is The “Bell Curve” or a Normal Distribution.

Answer 13

The Normal Distribution has:
1. mean = median = mode
2. symmetry about the center
3. 50% of values less than the mean
and 50% greater than the mean

4. Standard deviations: 68% of values are within
1 standard deviation of the mean
95% of values are within
2 standard deviations of the mean
99.7% of values are within
3 standard deviations of the mean

Question 14

“Standard Score”, “sigma” or “z-score”.

Answer 14

The number of standard deviations from the mean

z = (x − μ) / σ

z is the “z-score” (Standard Score)
x is the value to be standardized
μ (‘mu”) is the mean
σ (“sigma”) is the standard deviation

Question 15

Correlation

Answer 15

When two sets of data are strongly linked together we say they have a High Correlation.

Correlation can have a value:
1 is a perfect positive correlation
0 is no correlation (the values don’t seem linked at all)
-1 is a perfect negative correlation

Question 16

“Correlation Is Not Causation” - 4 reasons why

Answer 16

What it really means is that a correlation does not prove one thing causes the other:

One thing might cause the other??
The other might cause the first to happen -simultaneous reverse dependence
They may be linked by a different thing -hidden 3rd variable
Or it could be random chance! -spurious

Question 17

Pearson’s Correlation formula

Answer 17

Step 1: Find the mean of x, and the mean of y
Step 2: Subtract the mean of x from every x value (call them “a”), and subtract the mean of y from every y value (call them “b”)
Step 3: Calculate: ab, a2 and b2 for every value
Step 4: Sum up ab, sum up a2 and sum up b2
Step 5: Divide the sum of ab by the square root of [(sum of a2) × (sum of b2)]

r = ( n Σ xy - Σx Σy ) / (square root( nΣx^2) - square root(Σx)^2) * (n Σy^2 - (Σy)^2) )

Question 18

Bayes Theorem

Answer 18

“AB AB AB” then remember to group it like: “AB = A * BA / B”

P(A|B) = P(A) * P(B|A) / P(B)

Which tells us: how often A happens given that B happens, written P(A|B),

When we know: how often B happens given that A happens, written P(B|A)

and how likely A is on its own, written P(A)

and how likely B is on its own, written P(B)

Question 19

Bayes Theorem applied to false positives and false negatives

Answer 19

P(A|B) =
P(A)P(B|A) /
( P(A)P(B|A) + P(not A)P(B|not A) )

Tp / (tp + fp)

Question 20

Birthday statistics formula

Answer 20

chance of n= people having the same r= random choices

R! / (R^n (r-n)!)

Then 1 - The result of this equation

The closer that n comes to r, the closer that the probability of N choose R will have a match comes to 100%

Question 21

Confidence intervals (integrations of total) for a standard distribution

Answer 21

Conf   Interval	Z
68%   1.0
80%	1.282
85%	1.440
90%	1.645
95%	1.960
99%	2.576
99.5%	2.807
99.9%	3.291

Question 22

How to calculate our Confidence interval (AKA Margin of Error) if we know our chosen z value.

Answer 22

use that Z value in this formula for the Confidence Interval
X ± Z * (s/√n)
Where:
• X is the mean
• Z is the chosen Z-value from the table in previous card
• s is the standard deviation
• n is the number of observations

Question 23

What is the formula for the confidence interval

Answer 23

The Confidence Interval is based on Mean and Standard Deviation. Its formula is:

X ± Z s√n

Where:

X is the mean
Z is the Z-value from the table below
s is the standard deviation
n is the number of observations

Question 24

formula for Chi-Square:

Answer 24

Χ^2 = Σ { (O − E)^2 / E }

Σ means to sum up (see Sigma Notation)
O = each Observed (actual) value
E = each Expected value
So we calculate (O−E)2E for each pair of observed and expected values then sum them all up.

Question 25

What is the Chi square test

Answer 25

The chi square test is to establish ‘p’ or the probability of independence.
p<0.05 is considered dependent!
p > 0.05 is considered independent!

Important points before we get started:
• This test only works for discreet categorical data (data in categories), such as Gender {Men, Women} or color {Red, Yellow, Green, Blue} etc, but not numericaldata such as height or weight.
• The numbers must be large enough. Each entry must be 5 or more. In our example we have values such as 209, 282, etc, so we are good to go.

Question 26

Formula for linear regression or line of Best fit

Answer 26

Calculate Slope m:
m = (N * Σ(xy) − Σx Σy) / (N * Σ(x^2) − (Σx)^2)
(N is the number of points.)

Calculate Intercept b:
b = (Σy − m * Σx) / N

Assemble the equation of a line
y = mx + b

Question 27

How does linear regression work?

Answer 27

It works by making the total of the square of the errors as small as possible (that is why it is called “least squares”):

The straight line minimizes the sum of squared errors
So, when we square each of those errors and add them all up, the total is as small as possible.

You can imagine (but not accurately) each data point connected to a straight bar by springs:

Be careful! Least squares is sensitive to outliers. A strange value will pull the line towards it.

Question 28

Grouped Frequency Distribution definition

Answer 28

It is also possible to group the values of collected data. they are grouped in Xs

It is very useful when the scores have many different values. Or when we want to go from continuous to discreet data

Question 29

How to determine Group Size or class intervals for Grouped Frequencies

Answer 29

Now calculate an approximate group size, by dividing the range by how many groups you would like.

Then round that group size up to some simple value (like 2 instead of 1.83 or 5 instead of 4.26).

Question 30

Estimating the Mean from Grouped Data

Continuous to Discreet

Answer 30

We can estimate the Mean by using the midpoints. 
EG: The groups (51-55, 56-60, etc), also called class intervals, are of width 5. The midpoints are in the middle of each class: 53, 58, 63 and 68

Class Interval A / 2 = Midpoint A

&a-z (Mid a * N of Mid a) / N

Estimated Mean = Sum of (Midpoint × Frequency) / Sum of Frequency

Add all Midpoints times N of data points within that midpoint
Then divide that sum by N or All data points

Question 31

Grouped Data Frequency median

Answer 31

Median is value at n/2.
Estimated Median = L + [(n/2 − B) / G ] × w

where:

L is the lower class boundary of the group containing the median (This is tricky, it starts where the values would round up to the lowest possible number in that class interval)
n is the total number of values
B is the cumulative frequency of the groups before the median group
G is the frequency of the median group
w is the group width or class interval

Question 32

Estimating the Mode from Grouped Data

Answer 32

We can easily find the modal group (the group with the highest frequency),

Estimated Mode = L + (fm − fm-1) / (fm − fm-1 + fm − fm+1) × w
L + (B-A) / ( B-A + B-C) * W
where:

L is the lower class boundary of the modal group (This is tricky, it starts where the values would round up to the lowest possible number in that class interval)
A (or fm-1) is the frequency of the group before the modal group
B (fm) is the frequency of the modal group
C (fm+1) is the frequency of the group after the modal group
w is the group width

Question 33

What is a Random Variable

Answer 33

It is a value we give to observed events that we are uncertain of beforehand

1. A Random Variable is a set of possible values from a random experiment.
2. The set of possible values is called the Sample Space.
3. A Random Variable is given a capital letter, such as X or Z.
4. Random Variables can be discrete or continuous.

EG:
We have an experiment (such as tossing a coin)
We give values to each event
The set of values is a Random Variable

Question 34

Probability is defined as

Answer 34

How likely a random variable is to occur

Probability of an event happening = Number of ways it can happen / Total number of outcomes

P(X = x) = x / n
P(X = value) = probability of that value

(Notice the different uses of X and x:
X is the Random Variable “The sum of the scores on the two dice”.
x is a value that X can take.)

Question 35

in Probability, An Experiment is a

Answer 35

Experiment: a repeatable procedure with a set of possible results.

Question 36

in Probability, An Outcome is a

Answer 36

Outcome: A possible result of an experiment.

Question 37

In probability, A Sample Space is a

And What are Sample Points?

Answer 37

Sample Space: all the possible outcomes of an experiment.

The Sample Space is made up of Sample Points:
Sample Point: just one of the possible outcomes

Question 38

In Probability, an Event is

Answer 38

Event: one or more outcomes of an experiment

Question 39

In probability

Analog or Continuous is

Answer 39

Random Variables can be either Discrete or Continuous:

Continuous Data can take any value within a range (such as a person’s height)

Question 40

Digital or Discreet in Probability means

Answer 40

Random Variables can be either Discrete or Continuous:

Discrete Data can only take certain values (such as 1,2,3,4,5)

Question 41

Mean or Expected Value: μ

Answer 41

When we know the probability p of every value x we can calculate the Expected Value (Mean) of X:

μ = Σxp

Note: Σ is Sigma Notation, and means to sum up.

To calculate the Expected Value:

• multiply each value by its probability
• sum them up

If all chances are random or equal then
1/n Σx = Σxp

Question 42

Variance: Var(X)

Answer 42

The Variance is:

Var(X) = Σx^2p − μ^2

To calculate the Variance:

square each value and multiply by its probability
sum them up and we get Σx^2p
then subtract the square of the Expected Value μ^2

Question 43

Standard Deviation: σ

Answer 43

Standard Deviation is the square root of the Variance:

σ = √Var(X)

Var (X) = Σx^2p − μ^2

Question 44

Biased choices or weighted combinations without repetition

N choose weighted r

Answer 44

But what if the coins are biased (land more on one side than another) or choices are not 50/50.

= p^k (1-p)^(n-k)

Where

p is the probability of each choice we want
k is the the number of choices we want
n is the total number of choices

p = 0.7 (chance of chicken)
k = 2 (chicken choices)
n = 3 (total choices)
So we get:

= p^k (1-p)^(n-k) = 0.7^2 (1-0.7)^(3-2)

Question 45

The General Binomial Probability Formula

Answer 45

Now we know how to calculate how many:
n!/(k!(n-k)!)

And the probability of each:
p^k (1-p)^(n-k)

When multiplied together we get:
Probability of k out of n ways:
P(k out of n) = n!/(k!(n-k)!) * p^k (1-p)^(n-k)

Important Notes:

The trials are independent,
There are only two possible outcomes at each trial,
The probability of “success” at each trial is constant.