Scheda di revisione: Fundamentals of Probability and Independence

📋 Course Outline

1. Purpose of probability and conditional focus
2. Frequencies in contingency tables
3. Probabilistic vocabulary: experiments and events
4. Conditional probability definition and interpretation
5. Weighted probability trees and path rules
6. Total probability formula and tree inversion
7. Independence of events and product rule

📖 1. Purpose of probability and conditional focus

🔑 Key Concepts & Definitions

• Rationalizing chance : Probability is used to quantify the likelihood of outcomes produced by a random experiment.
• Random experiment : A random experiment is a procedure whose outcome cannot be predicted in advance.
• Conditional probabilities : Conditional probabilities are probabilities computed after restricting the situation using extra information.
• Contingency table : A contingency table cross-tabulates two characteristics of a population using counts in each cell.

📝 Essential Points

• Probability originated from gambling problems such as card and dice games.
• Modern probability is used in many fields like finance, insurance, medicine, and accident analysis.
• From earlier studies, students learn general methods such as reading tables and building probability trees.
• This chapter introduces a new type of probability: probabilities conditioned on additional information.
• Conditional probability calculations require restricting the reference universe to a subset defined by the condition.

💡 Memory Hook

Chance → quantify; then add info → restrict universe → conditional probability.

📖 2. Frequencies in contingency tables

🔑 Key Concepts & Definitions

• Marginal frequency : A marginal frequency is the proportion of a population having a given value of one characteristic.
• Conditional frequency : A conditional frequency is the proportion of one value of a characteristic among the individuals already having another value.
• Marginal count : A marginal count is the total number in a row or column of a contingency table.
• Conditional frequency notation : Conditional frequency of b1 among a1 is denoted f_{a1}(b1).

📝 Essential Points

• Marginal frequency of a1 equals the marginal count T1 divided by the total population size T.
• Conditional frequency f_{a1}(b1) equals the cell count for (a1,b1) divided by the marginal count for a1.
• In the example table, women frequency is 208/577 ≈ 0.36 (36%).
• In the example table, the frequency of men among those aged over 60 is 142/223 ≈ 64%.
• In the example table, the frequency of over-60 among men is 142/369 ≈ 38%, showing the direction matters.
• The two last quantities are conditional frequencies but they must not be confused because the conditioning set differs.

💡 Memory Hook

Marginal: divide by T; Conditional: divide by the conditioning marginal (row/column you restrict to).

📖 3. Probabilistic vocabulary: experiments and events

🔑 Key Concepts & Definitions

• Random experiment : A random experiment is the object of study of a random phenomenon.
• Elementary outcome : An elementary event is an event containing exactly one possible outcome.
• Universe of the experiment : The universe Ω is the set of all possible outcomes of a random experiment.
• Event : An event is a set of outcomes from the universe.
• Outcomes (eventualities) : Outcomes are the possible results of a random experiment, typically denoted e_i.

📝 Essential Points

• The universe Ω is the set of the n possible outcomes of the experiment.
• Outcomes are often written as e_i, and events are written using braces with commas.
• An event is any subset of outcomes, so it can contain one or many outcomes.
• For a six-sided die, the universe is Ω={1,2,3,4,5,6}.
• For the die, the event “even number” is A={2,4,6}.
• The event “get a six” is B={6} and is an elementary event.

💡 Memory Hook

Ω = all outcomes; event = subset of Ω; elementary event = subset with 1 outcome.

📖 4. Conditional probability definition and interpretation

🔑 Key Concepts & Definitions

• Conditional probability : Conditional probability PB(A) is the probability of A given that B has occurred.
• Intersection event : The intersection A∩B is the event that both A and B occur together.
• Restriction of the universe : Computing PB(A) corresponds to restricting the reference universe to the outcomes where B holds.
• Nonzero condition : The conditional probability PB(A) is defined only when P(B)≠0.

📝 Essential Points

• Conditional probability is defined by PB(A)=P(A∩B)/P(B) when P(B)≠0.
• Because 0≤P(A∩B)≤P(B), PB(A) always lies in [0,1].
• If P(B) and P(A) are nonzero, then P(A∩B)=PB(A)×P(B).
• Under the same nonzero assumption, P(A∩B)=PA(B)×P(A).
• Interpretation: in the example, PF(J) is computed by discarding men and keeping only women as the reference universe.
• Conditional probability uses extra information: the condition B is known to be true.

💡 Memory Hook

PB(A)=P(A∩B)/P(B): divide by the probability mass of the condition.

📖 5. Weighted probability trees and path rules

🔑 Key Concepts & Definitions

• Weighted tree : A weighted probability tree represents successive choices with probabilities attached to branches.
• Node reference universe : A branch probability is computed relative to the universe represented by the node you start from.
• Path probability : The probability of a path is the probability of the corresponding sequence of events along the tree.
• Tree rules : Tree rules specify how branch sums and path products determine probabilities.

📝 Essential Points

• On a weighted tree, the probabilities of branches leaving the same node sum to 1.
• The probability of a path equals the product of the branch probabilities along that path.
• In the example tree, the factor 1/104 is the probability of “less than 30 years” among women.
• The tree encodes conditional probabilities by placing the condition as the earlier stage.
• The product rule matches intersections: PB(A)×P(B)=P(A∩B).
• The sum rule matches partitioning by the condition: PB(A)+PB(Ā)=1 at a given node.

💡 Memory Hook

Tree: same-node sum =1; along-path product = intersection probability.

📖 6. Total probability formula and tree inversion

🔑 Key Concepts & Definitions

• Total probability formula : The total probability formula expresses P(A) as a sum of probabilities of A across a partition by B and not-B.
• Partition by a condition : A partition splits the universe into two disjoint cases such as B and its complement B̄.
• Tree inversion : Tree inversion rewrites a probability tree by swapping which events appear on the first and second stages.
• Disjoint paths : Disjoint paths correspond to mutually exclusive cases, so their probabilities add.

📝 Essential Points

• The total probability formula is P(A)=P(A∩B)+P(A∩B̄).
• Event A is the union of the two disjoint cases corresponding to the two paths through B and through B̄.
• The sum works because the cases are disjoint: outcomes counted in B and in B̄ cannot overlap.
• In the example, P(J)=P(F∩J)+P(F̄∩J) is computed using the tree values.
• Using the given numbers, P(J)=208/577×1/104 + 369/577×7/369 = 9/577.
• Tree inversion is used when P(B) is not explicitly present in the original tree, so the missing probability is obtained via the total probability relation.

💡 Memory Hook

Total probability: add the two disjoint path probabilities (through B and through B̄).

📖 7. Independence of events and product rule

🔑 Key Concepts & Definitions

• Independence : Two events are independent if knowing one does not change the probability of the other.
• Conditional probability criterion : Independence can be tested by comparing PB(A) with P(A).
• Product rule for independent events : For independent events, the probability of the intersection equals the product of their probabilities.
• Successive independent experiments : Successive experiments are independent when the result of one does not affect the others.

📝 Essential Points

• Independence definition: events A and B are independent if PA(B)=P(B).
• Equivalently (stated as a property), independence holds iff P(A∩B)=P(A)×P(B).
• From independence, P(A∩B)=PA(B)×P(A)=P(B)×P(A).
• Conversely, if P(A∩B)=P(A)×P(B), then PA(B)=P(A∩B)/P(A)=P(B).
• For independent successive experiments, the probability of a whole list of outcomes is the product of the elementary outcome probabilities.
• Example with three independent die rolls: P(421)=1/6×1/6×1/6=1/216, and this is not the same as rolling three dice simultaneously as an unordered combination.

💡 Memory Hook

Independence ⇒ intersection = product; successive independent steps ⇒ multiply along the tree.

📊 Synthesis Tables

Marginal vs conditional frequencies

Quantity	Denominator	What it measures
Marginal frequency	Total T	Proportion of a value in the whole population
Conditional frequency f_{a1}(b1)	Marginal count T1 for a1	Proportion of b1 inside the sub-population with a1

⚠️ Common Pitfalls & Confusions

1. Confusing conditional frequencies because the conditioning set changes the denominator (e.g., “men among over-60” vs “over-60 among men”).
2. Using the conditional probability formula when P(B)=0, which makes PB(A) undefined.
3. Forgetting that branch probabilities on a tree are relative to the universe at the node, not the original Ω.
4. Adding path probabilities that are not disjoint, or multiplying probabilities that do not correspond to a single path.
5. Thinking that P(421) from three sequential die rolls equals the probability of the unordered combination when rolling three dice simultaneously.

✅ Exam Checklist

1. Be able to compute marginal frequency from a contingency table using marginal count over total.
2. Be able to compute conditional frequency f_{a1}(b1) as the cell count over the marginal count for a1.
3. Be able to define Ω, outcomes e_i, events as subsets, and identify an elementary event.
4. Be able to compute conditional probability PB(A)=P(A∩B)/P(B) and interpret it as restricting the universe to B.
5. Be able to apply weighted tree rules: same-node branch probabilities sum to 1 and path probability is the product of branch probabilities.
6. Be able to apply the total probability formula P(A)=P(A∩B)+P(A∩B̄) to compute missing probabilities and support tree inversion.
7. Be able to test independence using PB(A)=P(B) and use the product rule P(A∩B)=P(A)×P(B) for independent events.
8. Be able to compute probabilities for successive independent experiments by multiplying probabilities of each elementary result along the sequence.