Stats3

Question 1: Probability of Students in Subjects

  • Given:

    • 40% of students study both maths and science.

    • 60% study maths.

  • Required:

    • Calculate the probability of a student studying science given they are studying maths.

  • Formulas used:

    • Conditional Probability: P(A|B) = P(A and B) / P(B)

  • The solution:

    • P(Math and Science) = 40% = 0.4

    • P(Math) = 60% = 0.6

    • Therefore, P(Science | Math) = P(Math and Science) / P(Math) = 0.4 / 0.6 = 2/3.

Question 2: Rolling a Die and Tossing a Coin

  • Perform actions: Roll a die and toss a coin.

  • Required:

    • Calculate the probability that the die shows an odd number and the coin shows heads.

  • Definitions:

    • Independent Events: When the outcome of one event doesn't affect the other.

  • The solution:

    • P(Die shows odd) = 3/6 (1, 3, 5 are odd)

    • P(Coin shows heads) = 1/2 (only one head)

    • Therefore, P(Odd and Heads) = P(Odd) * P(Heads) = (3/6) * (1/2) = 1/4.

Recap of Previous Concepts

Bayes' Theorem

  • Overview of Bayes' theorem discussed in previous class.

  • Application of Bayes' theorem to examples.

    • Formula: P(A|B) = P(B|A) * P(A) / P(B)

    • Explained the understanding of 'at least' and 'at most'.

    • "At least" means greater than or equal to; "At most" means less than or equal to.

Detailed Example using Bayes' Theorem

  • Example: A bag contains 3 coins (2 fair, 1 fake).

  • Task: Find probability that the coin chosen was fake given that it showed heads.

  • Solution steps:

    1. Identify outcomes:

      • P(Head|Fake) = 1 (always shows head)

      • P(Fake) = 1/3

      • P(Head) calculated using law of total probability.

    2. Tree Model used to calculate - P(Head) from both fair and fake coins.

    3. Final probability calculated as 1/2.

New Questions for Practice

Card Example:

  • There are 3 types of cards with different color distributions:

    • 1 card with red on both sides.

    • 1 card with black on both sides.

    • 1 card with red on one side, black on the other.

  • Query: If one side of the picked card is red, what is the probability it's the red-black card?

Advanced Problem on Email Detection

  • Scenario: Emails received are classified as spam with certain detection rates.

  • Given that an email was detected as spam, find the probability that it was not spam.

  • Relevant Probabilities:

    • 50% chance of receiving a spam email.

    • 99% accuracy in detecting actual spam.

    • 5% chance of a non-spam email being incorrectly flagged.

  • Application of Bayes' theorem to calculate probability.

Addition of Probabilities

  • Concept clarification:

    • Probability of A or B: P(A or B) = P(A) + P(B) - P(A and B).

  • Example using cards to demonstrate how to correctly calculate probabilities when events overlap.