4.1, 4.2 MATH 11

Introduction to Probability

• Chapter Problem: Drug Testing of Job Applicants

  • Context: Approximately 85% of US companies test employees/job applicants for drug use.

  • Accuracy Challenges: Medical tests, including drug tests, can yield wrong results, characterized as:

    • False Positive: Indicates drug use when there is none.

    • False Negative: Misses drug use when it is present.

Understanding the Test Results

• Table Analysis:

  • Subjects who use drugs:

    • True Positive: 45 tested positive.

    • False Negative: 5 tested negative despite drug use.

  • Subjects who do not use drugs:

    • False Positive: 25 tested positive despite no drug use.

    • True Negative: 480 tested negative correctly.

• Implications: This table will be referenced frequently in understanding probability concepts.

Concepts of Probability

• **Basic Definitions: **

  • Event: Collection of outcomes of a procedure.

  • Simple Event: An outcome that cannot be divided further.

  • Sample Space: All possible simple events.

Example: Births and Gender Selection

• Single Birth Events: E.g., outcomes can be 'girl' or 'boy.'

• Three Births Event: E.g., outcomes can include combinations like 'two boys and one girl.'

Calculating Probability

• Example Calculation for Three Births:

  • Events like 'two boys and one girl' can occur via:

    • Boy, Boy, Girl (BBG)

    • Boy, Girl, Boy (BGB)

    • Girl, Boy, Boy (GBB)

  • The event 'two boys and one girl' is not simple since it decomposes into simpler events.

Coin Flip Probability

• Flipping Two Coins: Mathematically, it is similar to flipping one coin twice.

• Sample Space for Two Coins:

  • HH (Heads, Heads), HT (Heads, Tails), TH (Tails, Heads), TT (Tails, Tails)

• Probability Calculation: For example, if calculating the chance of one tail:

  • Two outcomes yield one tail (HT, TH) out of four total outcomes.

  • Probability = 2/4 = 0.5.

Probability Types

• Relative Frequency Approximation of Probability:

  • Based on observed outcomes.

  • Example: Flipping a coin 10 times with results recorded to find empirical probabilities.

• Classical Probability:

  • Based on equally likely outcomes.

• Subjective Probability:

  • Based on personal judgement or opinion, not calculable mathematically.

Example of Classical Probability

• Three Children Scenario:

  • What’s the probability of having all three children be the same gender?

  • Calculation involves counting outcomes in sample space (8 total outcomes) that meet criteria (2 outcomes).

Understanding Complements

• Complement of an Event (A):

  • Denoted by ( \bar{A} )

  • Consists of all outcomes where event A does not occur.

Mathematical Relationship

• Probability of A: ( P(A) )

• Complement Rule: ( P(A) + P(\bar{A}) = 1 )

  • If ( P(A) = 0.75 ) then ( P(\bar{A}) = 0.25 )

Odds and Their Calculation

• Odds Against Winning:

  • Defined as probability of the complement divided by the probability of the event.

• Payout Odds:

  • The ratio of net profit to amount bet.

  • Example with roulette wheel: Bet $5 on outcome.

Example: Roulette Probability

• Single slot choice outcome:

  • Bet on 13 with calculated odds of 1/38, payout odds of 35/1 - shows house edge.

Drug Testing Table Analysis

• Finding Probabilities:

  • How to derive answers from the provided testing table.

  • Important to understand what is being asked and how totals are formed using sample spaces.

Additional Rule for Probability

• Addition Rule for Disjoint Events:

  • ( P(A ∪ B) = P(A) + P(B) - P(A ∩ B) )

  • If A and B are disjoint: ( P(A ∩ B) = 0 )

Sample Space Illustrations

• Disjoint Events vs. Non-Disjoint Events:

  • Examples include distinct gender selections being disjoint, while king and red being non-disjoint (they can occur simultaneously).

Conclusion

• As probability concepts develop, understanding the application and meaning of each type becomes critical. This foundation will assist as we progress to more complex statistical methods in upcoming chapters.