1. Sixth lecture Concepts of Probabilities

2. Random Experiment Can be repeated (theoretically) an infinite number of times Has a well-defined set of possible outcomes. Result cannot be predetermined. Example 1: Selecting 20 people at random and counting how many people are left-handed. Tossing a coin until the first tail appears.

3. How can we describe random experiments using a mathematical model? We use three building blocks: a sample space, a set of events and probability. Definition 1 (Sample Space): The set of all possible outcomes of a statistical experiments is called the sample space and represented by the symbol S. Example 2: Select 20 people at random and count # of left- handers S= {0, 1, 2, …,20}

4. Example 3: Consider the experiment of tossing a die. If we are interested in the number that shows on the top face, describe the sample space? S= {1, 2, 3, 4, 5, 6} If we are interested only in whether the number is even or odd, the sample space is simply S= {even, odd}

5. Definition 2 (An event is) : The subset of a sample space. Donated by a capital letter. Example 4: Tossing two dice and A is the event that the sum of the two faces is ≥ 10. S= {(i, j), 1≤ i, j ≤ 6} A= {( 4, 6), (5, 6), (6, 6), (6, 4), (6, 5), (6,6)}.

6. The complement: The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol A ˊ.

7. The intersection The intersection of two events A and B, denoted by the symbol A ∩ B, is the event containing all elements that are common to A and B. A ∩ B

8. Definition 3: Two events A and B are mutually exclusive or disjoint, if A ∩ B= Ø, that is, if A and B have no elements in common. Definition 4: The union of the two event A and B denoted by the symbol A U B is the event containing all the elements that belong to A or B or both.

9. Example 5: (set operations) let S= {0, 1, 2, …, 7}, A= {0, 2, 6}, B= {1, 2} and C= {0, 6}. Find A ∩ B, A U B, B ∩ C, A ˋ. A ∩ B = {2} A U B = {0, 1, 2, 6} B ∩ C= Ø A ˋ = {1, 3, 4, 5, 7}

10. Some important results related to the set operation: A ∩ Ø = Ø. A U Ø = A. A ∩ A ˊ = Ø. A U A ˊ = S. S ˊ = Ø. Ø ˊ = S. (A ˊ ) ˊ = A. (A ∩ B) ˊ = A ˊ U B ˊ (A U B) ˊ = A ˊ ∩ B ˊ

11. Probability functions Want to assign a probability to an experiment's outcome (and in general to events). Let A be an event defined on sample space S P(A) denoted the probability of A occurring P is the probability function Definition 5: The probability of an event A is the sum of the weights of all sample points in A. Such that Axiom 1: 0 ≤ P(A) ≤ 1,

12. Axiom 2: P(S)= 1 and P(Ø) = 0, Axiom 3: if A, B, C, …. Is a sequence of mutually exclusive events then P(A U B U C U …)= P(A) + P(B) + P(C)+ …. Example 6: A coin is tossed twice. What is the probability that at least one head occurs?

13. Solution : The sample space is S= {HH, HT, TT, TH} Let A be the event that at least a head occurs. If we assume the coin is balanced all outcomes would be equally likely to occur, therefore The probability for any element to occur is ¼ And P(A)= ¾.

14. Theorem 1: If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is P(A)= n/N

15. Example 7: Consider the experiment of tossing a die. What is the probability that we get an even number? Solution: The sample space is S= {1, 2, 3, 4, 5, 6} Let A be the event that we get an even # A={2,4,6}, because the outcomes all equally likely to happened, so we will be able to apply the previous theorem. N = 6, n= 3 therefore P(A) = 3/6=1/2.

16. Additive Rules Theorem 2: If A and B are two events, then P(A U B)= P(A) +P(B) – P(A ∩ B). Result 1: If A and B are mutually exclusive, then P(A U B)= P(A) +P(B) If A ₁, A ₂, A ₃, … A is a partition of sample space S then P(A ₁ U A ₂ U A ₃ U … U A )= P(A ₁ ) + P(A ₂ )+ P(A ₃ ) + … +P(A)=P(s)=1.

17. Theorem 3: If A and A ˊ are complementary events then P(A)+ P(A ˊ )= 1. Example 8: What is the probability of getting a total of 7 or 11 when a pair of fair dice are tossed?