Intro to Probabiltiy theory notes for Elements Class.pptx

Probability complementary events the complement of a

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Probability
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Complementary Events: The complement of A: Ǟ is the set of all sample points in the sample space that does not belong to event A . i.e. If A , is heads, then Ǟ is tails. Complement Rule: In words: probability of A complement = one – probability of A In algebra: P( ) = 1 – P(A) Ǟ 24
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Some Properties of Probability Laws Consider a probability law, and let A , B , and C be events. (a) If A ⊂ B , then P ( A ) P ( B ). (b) P ( A ∪ B ) = P ( A ) + P ( B ) P ( A ∩ B ). (c) P ( A ∪ B ) P ( A ) + P ( B ). (d) P ( A ∪ B ∪ C ) = P ( A ) + P ( A c ∩ B ) + P ( A c ∩ B c ∩ C ). 25
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26 CALCULATING THE PROBABILITY OF AN EVENT Example: The subjects in the study consisted of a sample of 75 men and 36 women. The subjects are a fairly representative sample of “typical” adult users who were neither in treatment nor in jail. Table 1 shows the life time frequency of cocaine use and the gender of these subjects. Suppose we pick a person at random from this sample. What is the probability that this person will be a male?
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27 We define the desired probability as the number of subjects with the characteristic of interest (male) divided by the total number of subjects. Symbolically P(M)= number of males/ total number of subjects = 75/111 = 0.6757
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28
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Properties of Conditional Probability The conditional probability of an event A , given an event B with P ( B ) > 0, is defined by P ( A | B ) = P ( A ∩ B )/ P ( B ) , and specifies a new (conditional) probability law on the same sample space Ω. In particular, all known properties of probability laws remain valid for conditional probability laws. 29
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Properties of Conditional Probability Conditional probabilities can also be viewed as a probability law on a new universe B , because all of the conditional probability is concentrated on B . In the case where the possible outcomes are finitely many and equally likely, we have P ( A | B )= [number of elements of A ∩ B] [number of elements of B] . 30
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31 Conditional Probability When probabilities are calculated with a subset of the total group as the denominator, the result is a conditional probability. Example:- Suppose we pick a subject at random from the 111 subjects and find that he is a male (M). What is the probability that this male will be one who has used cocaine 100 times or more during his lifetime? This is a conditional probability and is written as P(C|M) or The probability of a cocaine user given that he is male . The 75 males become the denominator of this conditional probability, and 25 the number of males who have used cocaine 100 times or more during their lifetime, becomes the numerator. Our desired probability, then is P(C| M) = 25/75 = 0.33
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32 Joint Probability The probability that a subject picked at a random from a group of subjects possesses two characteristics at the same time.
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