Statistics 2 LW1 - Probabilities

Probability

• Probability: The measure of the likelihood that an event will occur (also known as chance or likelihood).

• Probability is used everywhere, so it is important to know how to interpret it.

Problems in Probability

• Polls and predictions

Why were the polls wrong?

• Samples were not representative.

• Wrong questions were asked.

• Failure to understand Statistical Measurements.

• Margin of error.

Relation with Business

• It helps to make decisions.

  • Compute the probability of an outcome and prepare accordingly (e.g., insurance).

• It is used for drawing conclusions based on small amounts of data.

  • Often used in research (e.g., market research).

The Birthday Problem

• Suppose there are 23 people. What is the chance two people share the same birthday?

Probability Rules & Conventions

• Probability is given the symbol $P$. $P$ is always between 0 and 1 ( $0 ≤ P ≤ 1$ ). 0 = 0% chance; 1 = 100% chance.

Basic Formulas

• $P = \frac{\text{How many times a specific thing happens}}{\text{How many times all things happen}}$

• More formally: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$, where E means Event.

Example

• Chance of throwing a 6 on a dice:

• $P(\text{throwing a 6}) = \frac{1}{6} = 0.1667$

Multiplication & Addition Rules

• Multiplication Rule: Used when you want to know the probability of two or more things happening.

• Example

  • Chance of throwing a 6 AND then throwing a 4: $P(\text{throwing a 6}) * P(\text{throwing a 4}) = (1/6) * (1/6) = 1/36 = 0.0278$

• Formula: $P(A \text{ and } B) = P(A) * P(B)$

Exercise 1

• Research about Pokémon preferences: 70% like Pikachu, 20% pick Squirtle, 50% Charmander, 30% Bulbasaur.

• Question: Chance that a person does not like Pikachu and chose Squirtle?

Exercise 1 Answer

• $P(\text{person does not like Pikachu}) = 0.3$

• $P(\text{person chose Squirtle}) = 0.2$

• $P(\text{person does not like Pikachu AND chose Squirtle}) = 0.3 * 0.2 = 0.06$

Multiplication Rule & Dependence

• The Multiplication Rule only works if the events are completely independent.

• Being in one category does not influence the outcome of the other category at all!

• E.g. Liking (or not liking Pikachu) does not influence whether you chose Squirtle.

Multiplication Rule & Dependence Example

• Gender and Favorite Television Series are independent; however, preference for Bridgerton and Better Call Saul are not independent.

• P(a person is female AND prefers Better Call Saul) using multiplication rule:

  • $(50/100)*(35/100) = 0.1750$

  • In the table, there are 10 women who prefer Better Call Saul.

• Conclusion: always draw a table!

Addition Rule

• The Addition Rule is used when we want to know the probability either (or all) of several events occurs.

• Example: Chance of throwing a 6 OR a 4.

  • $P(\text{throwing a 6}) + P(\text{throwing a 4}) = 1/6 + 1/6 = 2/6 = 0.3333$

• Formula: $P(A \text{ or } B) = P(A) + P(B)$

Addition Rule

• The addition rule only works when events are Exclusive (they cannot happen at the same time).

• When Events are NOT exclusive we use:

  • $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Addition Rule Example

• What is the chance that a randomly picked person is Female OR likes the Office?

• Gender and Favorite Television Series are NOT exclusive.

• The chance that someone is Female OR likes the Office best (or both):

  • $50/100 + 30/100 – 15/100 = 0.65$

• $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Exercise 2

• Research at Arnhem Business School about the preference for type of social media (facebook, Instagram, TikTok).

• You asked 100 people in total, 60 teachers, 30 people preferred instagram, 10 teachers preferred instagram, 40 teachers and 10 students preferred facebook

• What is the chance that a randomly picked person is a student or preferred TikTok?

Exercise 2 Solution

• P(student OR TikTok) = 40/100 + 20/100 – 10/100 = 0.5

Complement Rule

• The complement rule is used when you want to find out the probability of something NOT happening.

• $P(\text{not } A) = 1 – P(A)$

  • This can simplify calculations.

The Birthday Problem

• Suppose there are 23 people. What is the chance two people share the same birthday?

• It's easier to approach using the complement rule

• What is the chance that 2 people DO NOT share the same birthday?

  • $\frac{365}{365} \text{ x } \frac{364}{365}$

• Chance of 23 people NOT sharing the same birthday:

  • $\frac{365}{365} \text{ x } \frac{364}{365} \text{ x } \frac{363}{365} \text{ x } \frac{362}{365} \text{ x } … \text{ x } \frac{343}{365} = 0.4927$

• Chance 2 people share the same birthday in a group of 23:

  • $1 - 0.4927 = 0.5073$

Conditional Probability

• Conditional probability is when we take a smaller part (a subset) from the total (e.g. only pick women) and calculate a probability within that subset.

• The notation looks like this: $P(A|B)$

  • Pronounced as: The probability of A given B (Here A is the event and B is the condition)

Conditional Probability Example

• What is the probability that someone likes TikTok given that they are a student?

• Or P(TikTok|Student)?

Conditional Probability Example Answer

• $P(\text{TikTok}|\text{Student}) = P(A|B)$

• There are 30 students who prefer TikTok (A).

• There are 60 students in total (B).

• Thus $P(A|B) = 30/60 = 0.5$

Conditional Probability: Why it’s important

• Conditional probability is a recurring problem in medicine.

• Malaria test is 95% accurate; people have a 1% chance of being affected with Malaria.

• If you have Malaria, there is a 5% chance that you are told you are not sick.

• If you do not have Malaria, there is a 5% chance you are told you are sick.

• If the test result is positive (the test says you have Malaria), what is the chance you are not actually sick?

• In other words: P(not sick|positive test result)?

Malaria Test Calculation

• Chance of having malaria and tested positive = 0.01*0.95

• Chance of having malaria and tested negative = 0.01*0.05

• Chance of not having malaria and tested positive = 0.99*0.05

• Chance of not having malaria and tested negative = 0.99*0.95

• Chance of testing positive = 0.010.95 + 0.990.05

• Chance of not having malaria | Chance of testing positive = 0.990.05 / (0.010.95 + 0.99*0.05) = 0.8390

Conditional Probability: Why it’s important Answer

• Let's say we ask 10,000 people

  • 100 will have Malaria, 95 told they have it, 5 told they don't.

  • 9900 will not have Malaria, 9405 told they don't have it, 495 told they do.

• Therefore, in total 495+95 = 590 people will be told they have Malaria

• The chance that you are told you have Malaria even though you don’t is therefore: 495/590 = 0.8390

Contingency Table

• Very useful for understanding conditional probabilities.

• Contingency table with counts gives a “cleaner” overview.

Probability and Keywords

• Each rule is generally tied to a specific keyword:

  • Multiplication Rule: AND

  • Addition Rule: OR

  • Conditional Rule: IF

  • Complement Rule: NOT

Probability and Keywords

• Multiplication Rule: AND, Events are independent, contingency table if not independent

• Addition Rule: OR, Events are exclusive, contingency table if non-exclusive, subtract $P(A \text{ and } B)$

• Conditional Rule: IF, Use contingency table or tree diagram

• Complement Rule: NOT

Dependent Independent Events | Flipping a coin Twice

• $P \text{(Twice heads)} = 0.5*0.5 = 0.25$

• $P \text{(First heads then tails)} = 0.5*0.5 = 0.25$

• $P \text{(First tails then heads)} = 0.5*0.5 = 0.25$

• $P \text{(Twice tails)} = 0.5*0.5 = 0.25$

Dependent & Independent occurrences

• Events are independent when the outcome of Event A does not influence the outcomes of event B.

• What you had on your first coin flip does not influence what happens on the second flip.

• Sometimes it happens that event A does influence event B. Then the events are Dependent.

Dependent Events

• Need to two students from a class (of 10) to help me out.

• In this class there are 6 girls and 4 boys

• The chance that the first person I pick is a girl = 6/10

• The chance that the second person I pick is also a girl = 5/9

Dependent Events: Tree Diagram

• $P \text{(Two girls)} = (6/10) * (5/9) = 0.3333$

• $P \text{(First girl then boy)} = (6/10) * (4/9) = 0.2667$

• $P \text{(First boy then girl)} = (4/10)*(6/9) = 0.2667$

• $P \text{(Twice boy)} = (4/10)*(3/9) = 0.1333$