Probability Concepts and Calculation Techniques

  • Introduction to Probability

    • Definition of probability and importance of notation.

    • All possible outcomes must sum to 1 (e.g., tossing a fair coin).

    • Fair coin example:

    • Outcomes: Heads (H) and Tails (T).

    • Probabilities: P(H) = 0.5, P(T) = 0.5.

  • Defining Events and Notation

    • Event of interest represented by notation (A).

    • For example, let A be the event that top side of coin is heads.

    • Probability of event A denoted as P(A).

    • Example Probability of hearing impairments: P(A) = 0.28.

    • Notation for complementary events:

    • Not A (denoted Å): P(¬A) = 0.72 for the case of no hearing impairment.

  • Contingency Tables

    • Importance of contingency tables in organizing probability data.

    • Example scenario:

    • Two categorical variables: Colour (Normal/Blue) & Hearing (Normal/Impaired).

  • Filling Out the Contingency Table

    • Ensure total probabilities sum to 1.

    • Data facts to populate the table:

    • 28% of donations: hearing impaired.

    • 11% of donations: blue-eyed.

    • 5%: both blue-eyed and hearing impaired.

    • Use the law of total probability to fill in the table.

  • Law of Total Probability

    • Fundamental principle in working with probabilities related to multiple events.

    • Example calculation: To find P(Normal Hearing, Blue-eyed), rearrange equation to involve existing values.

    • Find probabilities in various cells using conditional probabilities and the total probability law.

  • Complementarity

    • Complementary events (A and ¬A) must sum to 1:

    • Example: Normal eye colour = 1 - P(Blue).

    • Calculate the required values for both variables (eye colour & hearing).

  • Conditional Probability

    • Definition: The probability of an event given that another event has occurred.

    • Example: P(Hearing Impaired | Blue-eyed) calculation.

  • Independence of Events

    • Distinguish between independent and dependent events.

    • Independence definition: P(A & B) = P(A) * P(B).

    • Examining whether eye colour and hearing impairment are independent.

    • Result: They are not independent; different probabilities result when calculating using different methods.

  • Applications of Probability

    • Practical implementations in fields such as mining.

    • Risk assessments based on independent events (e.g., equipment failures).

    • Use of complementary probabilities for practical evaluations.

  • Conclusion

    • Probability is a versatile tool used for understanding events in various disciplines.

    • Thorough understanding of independence, conditional probability, and how to manipulate probabilities is essential for accurate data analysis.