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Probability: Conditional, Independence & Rules | Examples, Trends

An overview of conditional probability, independence, and the multiplication and addition rules in probability. Includes various examples to illustrate concepts.

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Probability: Conditional, Independence & Rules | Examples, Trends

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  1. More Probability (Dr. Monticino)

  2. Overview • Conditional probability and independence • Multiplication and addition rules • Examples, examples, examples

  3. Conditional Probability • Recall, that conditional probability provides a formal way for conditioning probabilities based on new information • In particular, • P(A | B) = P(A  B)/P(B) • P(A B) = P(A | B)  P(B)

  4. Independence • Intuitively, two events are independent if information that one occurred does not affect the probability that the other occurred • More formally, A and B are independent if • P(A | B) = P(A) • P(B | A) = P(B) • P(A  B) = P(A)P(B)

  5. Multiplication Rule • The probability of the intersection of two events equals the probability of the first multiplied by the probability of the second given that the first event has happened • P(A B) = P(A | B)  P(B) • If the two events are independent, then • P(A B) = P(A)  P(B)

  6. Addition Rule • If two events are disjoint, then the probability that at least one happens is the sum of the probabilities of each event • P(A  B) = P(A) + P(B) • 1= P() = P(A  Ac) = P(A) + P(Ac) • So, P(A) =1 - P(Ac) • Thus, you can only add two probabilities if the events are mutually exclusive.

  7. Examples • Draw two cards are drawn from a well-shuffled deck of cards • What is the probability that the first card is either a queen or a king? • Find the probability that the first is a queen and the second is a king • Without replacement • With replacement • Find the probability that the second card is a king? • Probability of drawing queen and king

  8. Examples • A coin is tossed three times • What is the probability of getting no heads? • What is the probability of getting exactly two heads? • What is the probability of getting at least one head? • Suppose that a coin is tossed 50 times • Given that the first 49 tosses come up tails, what is the probability that the 50th toss is a head? • What is the probability of getting 50 tails in a row?

  9. Examples • Paradox of the Chevalier de Mere • The birthday problem (Dr. Monticino)

  10. Birthday Probabilities

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