Chapter 11 - 1

Chapter 11: Introducing Probability

• Key Concepts

  • The idea of probability

  • Probability models

  • Probability rules

  • Finite probability models

  • Continuous probability models

  • Random variables

  • Personal probability

Chance Behavior

• Chance behavior is unpredictable in the short term but shows regular patterns in the long term.

• Understanding chance behavior helps to trust randomization in selecting samples or assigning treatments in experiments.

• Relative Frequency:

  • Defined as the number of times an outcome occurs divided by the total number of trials.

  • Higher relative frequency implies that the outcome occurs more often.

Randomness and Probability

• A phenomenon is random if outcomes are uncertain but show a regular distribution in many repetitions.

• The probability of an outcome is the ratio of the number of times it occurs to the total number of trials performed.

• Example (Coin Flipping):

  • Cannot predict toss outcome but can estimate the probability of heads over many flips.

  • Probability is empirically derived.

Historical Coin Tossing Examples

• Count Buffon (4040 tosses): 2048 heads (0.5069)

• Karl Pearson (24,000 tosses): 12012 heads (0.5005)

• John Kerrich (10,000 tosses during WWII): 5067 heads (0.5067)

Coin Tossing

• Over many tosses, the probability of heads is approximately 0.5.

• Results of single tosses are random, but multiple tosses produce predictable results, assuming independence.

100 Tosses of a Fair Die

• Represents randomized outcomes over die rolls.

• Observations of outcomes may vary but should stabilize around expected probabilities over many trials.

Probability Models

• Describe chance behavior with:

  • A list of possible outcomes.

  • A probability for each outcome.

• Sample Space (S): Set of all possible outcomes.

• Events (A): Collections of sample points (subsets of the sample space).

• A probability model consists of a sample space and a way to assign probabilities.

Coin Example

• Sample Space for Flipping a Fair Coin: S = {H, T}

• Probabilities: P(H) = 1/2, P(T) = 1/2.

Tree Diagrams and the Fundamental Counting Principle

• Tree Diagram: Visual representation of all possible outcomes.

• Fundamental Counting Principle: If one task can be done in m ways and another in n ways, they can be done in m × n ways.

Independent Trials in Coin Tossing

• Outcomes of independent trials do not affect each other.

• Probability of sequences remains consistent across different trials.

• Example: Observing 2 heads among 5 tosses versus a specific sequence (like TTHTH) has an equal probability.

Practice Questions

• Describe the sample space for rolling two six-sided dice, determining probabilities for sums of 2, 5, and 7.

Probability Rules

1. Probabilities range from 0 to 1.

2. Probability of an event is the sum of probabilities of its sample points.

3. Probability of the empty set is 0.

4. Sum of probabilities for events in a sample space equals 1.

5. Complement Rule: P(AC) = 1 – P(A).

6. For disjoint events A and B: P(A or B) = P(A) + P(B).

Example of Probability Distribution

• Canadian's mother tongue survey example with probabilities assigning to various languages.

Random Variables

• Definition: Numeric outcomes of random phenomena, typically denoted by capital letters (X, Y).

• Types:

  • Discrete: Countable number of outcomes (e.g., coin flips, die rolls).

  • Continuous: Infinite possible outcomes (e.g., temperature).

Probabilities in Discrete Models

• Assign probabilities to each outcome: sum equals 1.

• Generally depicted as relative frequency histograms.

• Example: Rolling dice results.

Finite Probability Models

• Sample space for heads in multiple tosses of a coin described via tables of probabilities.

• Practice exercises involve analyzing probabilities across various outcomes of finite trials.

Continuous Probability Models

• Cannot assign probabilities to individual outcomes.

  • Probabilities are represented as areas under density curves.

• The area under the curve equals 1 over the entire sample space.

Normal Distributions

• Probability models like Normal curves aligned with data distributions.

• Example: Heights of young women approximating a Normal distribution.

Practice Examples

• Pregnancies and skull sizes: use conception-to-birth data and measurements to assess probabilities and distributions in context.