Study Notes on Conditional Probability

Definition: Conditional probability refers to the probability of event A occurring given that event B has already occurred. It is denoted as P(A|B).
Example Scenario: Deciding between two cars where initial probability of choosing either car is 50%. After analyzing reviews, you find:
- Car 1 has a 40% repair rate in the first year.
- Car 2 has a 10% repair rate in the first year.
After considering the information, the probabilities change from equal likelihood (50%) based on additional data about the cars' performance.

New Sample Space: When event B occurs, it constrains the sample space to just the outcomes relevant to event B, which is referred to as the restricted sample space (R).
If you use R consistently when calculating conditional probabilities, this simplifies the process significantly.

The notation used is P(A|B), where:
- A = event of interest.
- B = the condition that has already occurred.

Case 1: What is the probability of getting a sum of 5, given that the first die is a 2?
- Restricted Sample Space: R = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)}
- Outcomes leading to a sum of 5: {(2,3)}
Case 2: What is the probability of getting a sum of 7, given that the first die is a 4?
- Restricted Sample Space: R = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)}
- Outcomes leading to a sum of 7: {(4,3)}
Case 3: What is the probability of the second die being a 2, given that the sum is 9?
- Restricted Sample Space: R = {(3,6), (4,5), (5,4), (6,3)}
- No outcomes leading to the second die being a 2.
Case 4: What is the probability of getting a Spade, given that the card is a Jack?
- Restricted Sample Space: R = {JS, JC, JD, JH}
- Outcomes leading to getting a Spade: {JS}.
Case 5: What is the probability of getting an Ace, given that the card is a Queen?
- Restricted Sample Space: R = {QS, QC, QD, QH}
- No outcomes leading to getting an Ace.

Independent Events: Two events A and B are independent if P(A|B) = P(A), meaning the occurrence of B does not affect A.
Dependent Events: Two events are dependent if P(A|B) P(A), indicating that knowing B has occurred impacts the likelihood of A occurring.

Dice Example:
- Events: Sum of 7 and first die being a 4 are independent.
- Events: Sum of 5 and first die being a 2 are dependent.
Cards Example:
- Events: Picking a Jack and picking a Spade are independent.
- Events: Picking a Heart and picking a Red card are dependent.
Children’s Births: The gender of one child does not affect the gender of another child; events are independent.
Coin Flips: The outcome of one flip does not affect subsequent flips; thus probability remains 0.5 regardless of past flips.

When calculating the joint probability of two dependent events:
- $P(A ext{ and } B) = P(A) imes P(B|A)$
For independent events: $P(A ext{ and } B) = P(A) imes P(B)$

Statistical data and calculations show the relationship between income and leprosy incidence tracked by WHO.
Key Findings:
- Probability of a person with leprosy being from the Americas is calculated by: $P(Americas) = \frac{36817}{222545} \approx 0.165$
- Probability of being from a high-income country: $P(High Income) = \frac{264}{222545} \approx 0.001$
- Joint probability for both conditions can be derived to analyze dependencies.

Understanding conditional probabilities and event dependencies is crucial for accurate statistical analysis and real-world decision-making, such as assessing risks, making choices in game theory, or strategic planning in various fields such as healthcare, economics, etc.

Evaluate the independence of various events.
Calculate probabilities based on the scenarios given.
Practice problems involving dice and cards to reinforce understanding of conditional probabilities and event independence.