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Sixth lecture Concepts of Probabilities 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. 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 lefthanders S= {0, 1, 2, … ,20} 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} 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)}. 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 Aˊ. 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 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. • 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, Aˋ. • A ∩ B = {2} • A U B = {0, 1, 2, 6} • B ∩ C= Ø • Aˋ= {1, 3, 4, 5, 7} Some important results related to the set operation: • A ∩ Ø = Ø. • A U Ø = A. • A ∩ Aˊ = Ø. • A U Aˊ = S. • Sˊ = Ø. • Ø ˊ= S. • (Aˊ)ˊ = A. • (A ∩ B)ˊ = Aˊ U Bˊ • (A U B)ˊ = Aˊ ∩ Bˊ 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, 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? 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. 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 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. 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 A₁, A₂, A₃, … Aᵣ is a partition of sample space S then P(A₁ U A₂ U A₃U … U Aᵣ )= P(A₁) + P(A₂)+ P(A₃) + … +P(Aᵣ)=P(s)=1. Theorem 3: If A and Aˊ are complementary events then P(A)+ P(Aˊ)= 1. Example 8: What is the probability of getting a total of 7 or 11 when a pair of fair dice are tossed?