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Scientific Inquiry SCI 105.020

Scientific Inquiry SCI 105.020. Probability & Normal Distribution. Probability. The basic idea of probability is to express the chance that an event occur at random in the long run . How many heads/tails you will see when tossing a coin a hundred times?

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Scientific Inquiry SCI 105.020

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  1. Scientific Inquiry SCI 105.020 Probability & Normal Distribution

  2. Probability • The basic idea of probability is to express the chance that an event occur at randomin the long run. • How many heads/tails you will see when tossing a coin a hundred times? • How likely is one candidate win a certain state in the general election? • How many smokers will die on cancer in the next decade?

  3. Basic Concepts • Event (denoted as A, B, or C): a result or collection of results of a particular experiment. • Sample Space (denoted as S): the collection of all possible outcomes of a particular experiment. • Probability: the likelihood that an event occurs in a particular experiment the long run • Examples of Sample Spaces: rolling dices • One dice: S = {1,2,3,4,5,6} • Two dices: S = {(1,1),(1,2),...,(6,6)} • Two dices + counting the total: S = {2,3,4,...,12}

  4. Probability • In a classical probabilistic model, we usually assume all outcomes of an experiment is equally likely, the probability of event A can be defined asP(A) = (# ways the outcomes in A can occur)/ (total # outcomes in the sample space) • Examples: • Rolling one diceand outcome is 2 or 3: P(A) = 2/6 = 1/3 • Drawing a card from a deck and out come is a king: P(A) = 4/52 = 1/13

  5. Addition Rule • Suppose that we have two events, A and B. Then, the addition rule states that P(A or B) = P(A) + P(B) – P(A and B) • Examples: • A and B are independent: • Drawing one card from a deck: a king or queen P(K or Q) = P(K) + P(Q) – P(K and Q) = 4/52 + 4/52 – 0 = 8/52 • A and B are independent: • Drawing one card from a deck: a king or spade P(K or Q) = P(K) + P(S) – P(K and S) = 4/52 + 13/52 – 1/52 = 16/52

  6. Probability Distribution • Random variable: a function whose value for each outcome of a procedure is determined by chance • Probability distribution: a graph, table, or function that gives the probability associated with each possible output of a random variable • Example: drawing 3 marbles, w/o replacement, from a bag with 5 black and 6 white marbles in it. # black marbles out of the 3 draws

  7. Probability Distribution: Properties • The area under the graph or bar chart must sum to a total of 1 (That is 100%) • For any particular value of the random variable x, the probability P(x) must fall between 0 and 1, inclusive. (That is, between 0 to 100%)

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