# Probability

##### by John T 06/24/2016

# Probability

by John T# 1. Our Big Picture Use sample statistics to make inferences or estimates about population parameters

# 2. Foundations

## 2.1. Statistics = science of planning studies and experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions based on the data

## 2.2. Data = collections of observations

### 2.2.1. Sample

2.2.1.1. Statistic

### 2.2.2. Population

2.2.2.1. Parameter

## 2.3. Variable = attribute that describes a person, place, thing, or idea, which can vary from one entity than another

### 2.3.1. Quantitative

### 2.3.2. Qualitative

## 2.4. Experiment vs. Observation Study

## 2.5. Statistical Significance vs. Practical Significance

# 3. Basic Concepts

## 3.1. Notation

### 3.1.1. P = probability (of)

3.1.1.1. Between 0 and 1

### 3.1.2. A, B, C... = events

### 3.1.3. P(A) = probability of event A

### 3.1.4. P(A bar) = probability of the complement of event A

## 3.2. Event = any outcome of an experiment

### 3.2.1. Simple event

3.2.1.1. Cannot be broken down

### 3.2.2. Compound event

3.2.2.1. Combines 2 or more simple events

## 3.3. Sample space = set of all possible outcomes

## 3.4. Simulation = a process that behaves the same as a procedure, so that results are similar

## 3.5. Measures of Test Reliability

### 3.5.1. Test sensitivity = probability of a true positive

### 3.5.2. Test specificity = probability of a true negative

## 3.6. An Event and Its Complement

### 3.6.1. Event A = event A occurs

3.6.1.1. (Event)

### 3.6.2. Event A bar = all outcomes in which event A does not occur

3.6.2.1. (Complement)

## 3.7. Rare event rule = if, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct

## 3.8. Counting

### 3.8.1. Permutations

3.8.1.1. Order matters

### 3.8.2. Combinations

3.8.2.1. Order doesn't matter

# 4. Approaches to Probability

## 4.1. Relative Frequency Approximation of Probability = results from observation of trials

### 4.1.1. Empirical

### 4.1.2. P(E) = # of observed trials with E / total # of outcomes

### 4.1.3. Law of Large Numbers = as the number of trials increases in an experiment, the relative frequency probability becomes closer to the theoretical probability

## 4.2. Classical Approach to Probability = based on sample space

### 4.2.1. Theoretical

### 4.2.2. P(E) = # of outcomes with E / # of outcomes in sample space

## 4.3. Subjective Probability = based on opinions

### 4.3.1. Subjective!

# 5. Solving Problems

## 5.1. Preliminary Questions

### 5.1.1. Can two events happen together?

5.1.1.1. Yes

5.1.1.1.1. The events are not disjoint / mutually exclusive

5.1.1.1.2. Not dependent

5.1.1.2. No

5.1.1.2.1. They are disjoint / mutually exclusive

5.1.1.2.2. Dependent

### 5.1.2. Does the probability of one event occurring affect the probability of another event occurring?

5.1.2.1. Yes

5.1.2.1.1. The events are dependent

5.1.2.2. No

5.1.2.2.1. The events are independent

### 5.1.3. Are you sampling with replacement?

5.1.3.1. Yes

5.1.3.1.1. The total number of possible outcomes stays the same

5.1.3.2. No

5.1.3.2.1. The total number of possible outcomes decreases

### 5.1.4. Does the order matter when counting?

5.1.4.1. Yes

5.1.4.1.1. Use permutations

5.1.4.2. No

5.1.4.2.1. Use combinations

## 5.2. Keywords

### 5.2.1. Addition Rule

5.2.1.1. OR

5.2.1.2. ALSO

### 5.2.2. Multiplication Rule

5.2.2.1. AND

5.2.2.2. ALL

5.2.2.3. EVERY

5.2.2.4. ENTIRE

### 5.2.3. Conditional Probability

5.2.3.1. IF

5.2.3.2. GIVEN

5.2.3.3. ASSUMING

### 5.2.4. Complement

5.2.4.1. NOT

### 5.2.5. Permutations

5.2.5.1. ORDER

5.2.5.2. SEQUENCE

5.2.5.3. ARRANGEMENT

5.2.5.4. SELECT

### 5.2.6. COMBINATIONS

5.2.6.1. SELECT

## 5.3. Specific Examples

### 5.3.1. Rule of Complementary Events

5.3.1.1. P(A) + P(A bar) = 1

5.3.1.2. P(A) = 1 - P(A bar)

5.3.1.3. Ex. Find the probability of finding an M&M is that is not blue in a sample of regular M&M's

### 5.3.2. Addition Rule

5.3.2.1. Mutually Exclusive Events

5.3.2.1.1. P(A or B) = P(A) + P(B) - P(A and B)

5.3.2.1.2. P(A or B) = P(A) + P(B)

5.3.2.1.3. Ex. Find the probability of rolling a 5 or a 6 with a dice

5.3.2.2. Non-Mutually Exclusive Events

5.3.2.2.1. P(A or B) = P(A) + P(B) - P(A and B)

5.3.2.2.2. Ex. Given "the probability that a child eats dinner alone or plays outside everyday", find the probability that a child eats dinner alone AND plays outside everyday"

### 5.3.3. Multiplication Rule

5.3.3.1. Dependent Events

5.3.3.1.1. P(A and B) = P(A) X P(B|A)

5.3.3.1.2. Includes sampling without replacement

5.3.3.1.3. Ex. Find the probability of drawing a King and then an Ace from a deck of cards

5.3.3.2. Independent Events

5.3.3.2.1. P( A and B) = P(A) X P(B|A)

5.3.3.2.2. P(A and B) = P(A) x P(B)

5.3.3.2.3. Includes sampling with replacement

5.3.3.2.4. Ex. Find the probability of flipping a coin on heads 5 times

### 5.3.4. Combining Addition and Multiplication Rules

5.3.4.1. Ex. P(A) X P(B|A) + P(Y) X P(Z|Y)

5.3.4.2. Ex. Given the probability of getting a disease with vaccine vs. not vaccine, where half of the population takes the vaccine and half doesn't, find the probability a random person will get the disease

5.3.4.2.1. P(z) = P(disease with vaccine)(.5) + P(disease without vaccine)(.5)

### 5.3.5. At Least One Rule

5.3.5.1. At least one = one or more = not none = not zero of the same type

5.3.5.2. P(A bar) = 1 - P(A)

5.3.5.2.1. P( at least 1) = 1 - P(none of that item)

5.3.5.3. Ex. Find the probability of getting at least 1 heads in 10 coin flips

5.3.5.3.1. Calculate the probability of getting no heads in 10 coin flips

### 5.3.6. Conditional Probability Rule

5.3.6.1. Conditional probability = calculating probabilities under some given conditions

5.3.6.2. P(A and B) = P(A) X P(B|A)

5.3.6.2.1. P(B|A) = P(A and B) / P(A)

5.3.6.3. The bigger number is on the bottom

5.3.6.4. Ex. Find the probability that a woman will have fraternal twins, given the fact she will have twins (along with a chart!)

5.3.6.4.1. P(fraternal twins) = # of fraternal twin births / # of total twin births

5.3.6.5. Ex. Find the probability that a bus will be late, given that it snows, if the probability of snow and the bus being late together is .023 and the possibility of snow tomorrow is .4

5.3.6.5.1. P(late|snow) = P(late and snow) / P(snow)

### 5.3.7. Counting

5.3.7.1. Multiplication Rule for Counting

5.3.7.1.1. Fundamental Counting Rule = the number of ways that two events can occur together if the first event can occur m ways and the second event can occur n ways is m X n ways

5.3.7.2. Factorial Rule

5.3.7.2.1. Factorial Rule = the number of different permutations of n different items when all n of them are selected is n!

5.3.7.2.2. Ex. How many ways can you arrange 3 plants in a row?

5.3.7.3. Permutations Rule

5.3.7.3.1. Permutation = the number of arrangements when many items are available but only some are selected

5.3.7.3.2. All Items are Different

5.3.7.3.3. Only Some Items are Different

5.3.7.4. Combinations Rule

5.3.7.4.1. Combination = the number of different combinations when many items are available and only some are selected without replacement and order doesn't matter

5.3.7.4.2. Ex. How many groups can you make out of 12 marbles where you only select 3?