Still in Ch2: 2.4 Conditional Probability 2.5 Independence

1. Still in Ch2: 2.4 Conditional Probability 2.5 Independence

2. Conditional Probability Section 2.4 Definition of conditional probability: For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by Which indicates that (the multiplication rule for conditional probability):

3. Conditional Probability and Independence Section 2.4-5 Example 2.24: Complex components are assembled in a plant that uses two different assembly lines (L1 and L2). L1 uses older equipment than L2, so it is somewhat slower and less reliable. Suppose on a given day L1 has assembled 8 components, of which 2 have been identified as defective (D) and 6 as nondefective (N), whereas L2 has produced 1 defective (D) and 9 nondefective (N) components.

4. Conditional Probability and Independence Section 2.4-5 Playing with the conditional probability definition! Look closely at A and A’, what do you see? Look closely at B in respect to A and A’, what do you see?

5. Conditional Probability and Independence Section 2.4-5 S A B A’

6. Conditional Probability and Independence Section 2.4-5 Which is called the Law of total probability.

7. Conditional Probability Section 2.4 Example 2.24: More questions to answer: Let us look at this again: What is the chance that the chosen component will be defective D? P(B|A) = 2/8 2 P(A) = 8/18 8 P(B|A) = 6/8 D B|A 6 1 L1 A P(A’) = 10/18 P(B|A’) = 1/10 10 N B’|A P(B|A’) = 9/10 D B|A’ 9 A’ L2 N B’|A’

8. Conditional Probability and Independence Section 2.4-5 Law of total probability, generalized: If A1, A2, …, Ak form a partition of the sample space (i.e. they are mutually exclusive and their union equals to S) then for any event B,

9. Conditional Probability and Independence Section 2.4-5 S A1 A2 A3 … Ak B

10. Conditional Probability and Independence Section 2.4-5 Playing with the conditional probability definition! Can be rewritten with the help of the multiplication rule and the law of total probability to be, Multiplication rule Law of total probability

11. Conditional Probability and Independence Section 2.4-5 Bayes Theorem: If A1, A2, …, Ak form a partition of the sample space (i.e. they are mutually exclusive and their union equals to S) then for any event B, such that P(B) > 0, For any Aj.

12. Conditional Probability Section 2.4 Example 2.24: More questions to answer: Given that the component was D, what is the chance that it was produced through line 2? P(B|A) = 2/8 2 P(A) = 8/18 8 P(B|A) = 6/8 D B|A 6 1 L1 A P(A’) = 10/18 P(B|A’) = 1/10 10 N B’|A P(B|A’) = 9/10 D B|A’ 9 A’ L2 N B’|A’

13. Conditional Probability and Independence Section 2.4-5 Why is Bayes Theorem imporant? Because some times all what we know is the conditional probabilities of some events.

14. Conditional Probability and Independence Section 2.4-5 • Example: Finding a fellow American in Paris: • English and American spellings are rigour and rigor, respectively. A man stays at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are American, what is the probability that the writer is an American?

15. Ch3: 3.1 Random Variables

16. Random Variables Section 3.1 An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty An outcome: is a result of an experiment. A sample space: is the set of all possible outcomes, S, of an experiment. An event: is a subset of the sample space. A simple event: is a subset of the sample space that has only one outcome. Probability: is a measure of the chance that an event might occur (before it really occurs).

17. Random Variables Section 3.1 Probability function on (simple) events in S S A number in [0,1] Snew Random variable; a function on outcomes in S Probability function on (simple) events in Snew

18. Random Variables Section 3.1 A Random Variable: is a function on the outcomes of an experiment; i.e. a function on outcomes in S. The input of this function is an outcome in S and the output is a number. Random variables are used to transform sample spaces to serve the goal of an experiment. A Variable: is a characteristic of a population of interest. An outcome is some kind of measurement of this characteristic.

19. Random Variables Section 3.1 We refer to a random variable by X, Y, Z, …, etc, indicating that the values of these variables are still not observed. Once we run an experiment and obtain an outcome, the observed realization of a random variable is referred to as x, y, z, …, etc. An observed realization of a random variable (or an outcome) is called a sample! (We already knew that, right?)

20. Random Variables Section 3.1 Examples: An experiment of flipping a coin once and observing a head: Probability distribution Sample space: {H,T} Random variable: X(H) = 1, X(T) = 0 or Probability distribution This is a discrete random variable! It’s sample space is finite.

21. Random Variables Section 3.1 Choosing an individual at random from last year’s class and checking if s/he received an A Sample space: {A, B, C, D, F} Probability distribution Random variable: X(A) = 1, X(B) = 0, X(C) = 0, X(D) = 0, X(F) = 0, or Probability distribution This is, also, a discrete random variable!

22. Random Variables Section 3.1 A random variable whose only values are 0 and 1 is called a Bernoulli random variable. If the P(X=1) = p and P(X=0) = 1 – p, then the Bernoulli probability distribution is, Bernoulli distribution Or, Or,

23. Random Variables Section 3.1 Rolling a dice twice and observing the sum Random variable X = sum of the faces on the two dice. X[(1,2)] = 1 + 2 = 3 Still discrete!

24. Random Variables Section 3.1 Flipping a coin until the first head shows up: S = {H, TH, TTH, TTTH, TTTTH, ….} Random variable X = Number of coin flips till termination of experiment. X(H) = 1, X(TH) = 2, … and Probability distribution Or, Discrete?

25. Random Variables Section 3.1 Observing the time until a red car passes through the main and sixth intersection. S = Random variable X = time till we observe a red car go through main and 6th. X(0) = 0, X(0.0001) = 0.0001 A one-to-one transformation of S. Probability distribution: Ch4 Continuous random variable! Sample space is continuous.

26. Random Variables Section 3.1 A discrete random variable is one that with a sample space that is finite or countably finite. (Countably finite => infinite yet can be matched to the integer line) A continuous random variable is one with a continuous sample space. More in Ch 4.