Probability_ MEDC 1114 WEEK 11_31Oct2024 (1)

Week 11: Introduction to Probability Theory

  • Instructor: Dr. L. Jeyaseelan, Professor of Biostatistics

Objectives

  • At the end of this tutorial, students should:

    • Distinguish between probability, proportion at risk, and percentage at risk.

    • State the laws of probabilities.

    • Apply operations of probabilities.

    • Calculate marginal, joint, and conditional probabilities.

    • Utilize the SPSS statistical package for calculations.

Key Questions Addressed

  • What is the probability of developing lung cancer for a smoker?

  • What is the probability of developing diabetes for an obese woman?

Sample Space

  • Defined as the set of all possible outcomes, denoted as S.

    • Example: Tuberculin skin test outcomes: S = {Positive, Negative, Uncertain}.

Events

  • An event is an outcome or a set of outcomes from a random phenomenon.

    • Example: If a tuberculin skin test shows TB, the event E = {Positive}.

Probability - Definition

  • Probability quantifies uncertainty and ranges from 0 to 1.

    • Values:

      • 0 indicates impossibility.

      • 1 indicates certainty.

Classical Definition of Probability

  • Probability (P) of an event A is given by the formula:

    • P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)

Example 1: Malnutrition Study

  • Sample of 50 children aged 5-9 years:

    • Calculated probability of malnutrition: P(Malnutrition) = 10/50 = 20%.

    • Sample Space: {Malnutrition, No Malnutrition}.

Empirical/Relative Frequency Probability

  • For experiments repeated n times with event E occurring m times:

    • Empirical probability = m/n.

Example 2: Vascular Headache Study

  • In a clinic, of 300 patients, 30 had vascular headaches:

    • Probability = P(Vascular Headache) = 30/300 = 0.10.

Axioms (Laws) of Probability

  1. Any probability is between 0 and 1. (0 ≤ p ≤ 1)

  2. Total probability of all possible outcomes must equal 1. (p(S)=1)

  3. Probability of an impossible event is 0.

  4. Complement rule: P(Ac) = 1 − P(A).

Properties of Probability

  • Certain events have a probability of 1 (e.g., P(Death)=1).

  • Events that cannot happen have a probability of 0 (e.g., P(Humans are Immortal) = 0).

Laws of Probability

Additive Property (Disjoint Events / Mutually Exclusive)

  • Events that cannot occur simultaneously (e.g., accidents on Saturday or Sunday):

    • P(Saturday or Sunday) = P(Saturday) + P(Sunday) = 0.02 + 0.03 = 0.05.

Addition Law of Probability (Not Mutually Exclusive)

  • For any two events A and B:

    • Pr(A or B) = Pr(A) + Pr(B) − Pr(A and B).

Example 3: STD Diagnosis

  • Two doctors diagnosed patients without sharing results:

    • Diagnoses for STD (Doctor A) = 0.1, Doctor B = 0.17, Overlap = 0.08.

  • Probability calculation: P(A or B) = P(A) + P(B) − P(A and B) = 0.19.

Multiplication Law of Probability

  • Probability of two independent events A and B occurring:

    • P(A and B) = P(A) x P(B).

Example of Independent Events

  • Considering a couple with two children, probability of both being girls:

    • P(Girl) = 1/2, P(Girl) for both = (1/2) * (1/2) = 1/4.

Example 4: Probability of Child Characteristics

  • Probability of a child being male and Rh positive:

    • P(Male) = 1/2, P(Rh+) = 9/10, P(Single Birth) = 79/80.

    • Combined probability = (1/2) * (9/10) * (79/80) = 711/1600 = 0.44.

Conditional Probability

  • Probability of two events occurring together:

    • Formula: P(A and B) = P(A) * P(B|A) = P(B) * P(A|B).

    • Example: Probability of lung cancer given smoking status.

Example 5: Lung Cancer Conditional Probability

  • Disease status probabilities:

    • Lung cancer (A): Yes = 0.12, No = 0.04 (total = 0.16).

    • Calculation: P(Lung Cancer | Smoker) = 0.12/0.16 = 0.75.

Marginal Probability

  • Also known as unconditional probability, not dependent on other events:

    • Example: Probability of lung cancer in a community.

Joint Probability

  • Probability of the intersection of two events A and B:

    • P(A ∩ B).

    • Example: If diagnosed with lung cancer and smoked, use probabilities to find joint probability.

Practicals and Exercises

SPSS Exercise

  • Utilize SPSS to reproduce findings from prior slides:

    1. Open SPSS Data: Question 1_HIV and TB.SAV

    2. Open SPSS Data: Question 2_Lung Ca and Smoke.SAV

Questions Relating to Color Blindness

  • Calculate probabilities associated with gender and color blindness:

    1. P(Color blind) - Marginal probability

    2. P(Male) - Marginal probability

    3. P(Color blind and male) - Joint probability

    4. P(Color blind | male) - Conditional Probability

    5. P(Color blind | woman) - Conditional Probability

Solutions for Color Blindness Questions

  1. P(Color blind) = 15/400 (Marginal)

  2. P(Male) = 200/400 (Marginal)

  3. P(Color blind and male) = 14/400 (Joint)

  4. P(Color blind | male) = 14/200 (Conditional)

  5. P(Color blind | woman) = 1/200 (Conditional)

Assignment

  • Complete Assignment 5.