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Probability – The Relative Frequency Definition. We define probability as a relative frequency. If some event A occurs f times out of N possible opportunities, we say that the probability of A is f divided by N. Probability experiments. We usually describe these as probability experiments.
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Probability – The Relative Frequency Definition • We define probability as a relative frequency. • If some event A occurs f times out of N possible opportunities, we say that the probability of A is f divided by N
Probability experiments • We usually describe these as probability experiments. • Thus: if an experiment is repeated enough, say N times, and if event A occurs f times, then the probability of A is
Intuitive Definition • Note that probability is an intuitive concept. • It is both analytic and empirical. • We theorize about probability analytically, and hence use formal proof. • We think about it and demonstrate it with real world observation.
Some Fundamentals • The probability of an impossible event must be zero (0.0) • The probability of a certain event must be 1.0 • Any and all probabilities must be between 0.0 and 1.0
Calculating Event probabilities • The probability that an event will occur is the sum of the sub-events that make comprise the event.
Composite Events • A composite event is one that comprised of combinations of other events • Such as • If either of two events occur • If both of two events occur
Intersection • Think in terms of Venn diagrams • Picture two sets A and B • What is the probability of being both A and or B? A C B
The Addition Rule • If A and B are two events, and their probabilities are denoted by P(A) and P(B), then the probability of either A or B or both is denoted by P(A or B). • P(A or B) = P(A) + P(B) - P(A and B)
The Addition Rule - a simple example • What is the probability of getting a head on either of two tosses of a coin? • P(A) = ½ • P(B) = ½ • P(A and B)=1/2 · ½ = ¼ • P(A or B) = P(A) + P(B) - P(A and B) • P(A or B) = ½ + ½ - ¼ = ¾
Mutually Exclusive events • If two events are mutually exclusive, they cannot both occur • (e.g. both a 2 and a 3 on a single role of a die) • The addition rule for mutually exclusive events is • P(A or B) = P(A) + P(B)
The Complement of an event • The complement of an event occurs when the event itself does not occur • P(A) + P(~A) = 1.0 • P(A) = 1.0 - P(~A)
Joint Probabilities • A joint probability is the probability that both A and B will occur – P(A and B)
