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Conditional Probability

Conditional Probability. 4.4. Problem. Create Venn Diagram. 17. 1. 45. 5. 2. 5. 25. C. Find. n(A) n(C) n(D) n(S) P(A) P(C) P(D) P(S) P(A') P(C') P(D') n(C U A) n(D U A) n(C U D) n(C ∩ A) n(D ∩ A) n(C ∩ D) P(A U C) P(D U A) P(D U C)

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Conditional Probability

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  1. Conditional Probability 4.4

  2. Problem Create Venn Diagram

  3. 17 1 45 5 2 5 25 C Find • n(A) n(C) n(D) n(S) • P(A) P(C) P(D) P(S) • P(A') P(C') P(D') • n(C U A) n(D U A) n(C U D) • n(C ∩ A) n(D ∩ A) n(C ∩ D) • P(A U C) P(D U A) P(D U C) • P(A ∩ C) P(D ∩ A) P(D ∩ C) • P(C' U A) P(A' U C) • P(C' ∩ A) P(A ∩ C’) • P(A U C)’ P(A ∩ C)’ • P(A ∩ C ∩ D) P(AUCUD) 10

  4. Conditional Probability In many situations, once more information becomes available, we are able to revise our estimates for the probability of further outcomes or events happening. For example, suppose you go out for lunch at the same place and time every Friday and you are served lunch within 15 minutes with probability 0.9. However, given that you notice that the restaurant is exceptionally busy, the probability of being served lunch within 15 minutes may reduce to 0.7. This is the conditional probability of being served lunch within 15 minutes given that the restaurant is exceptionally busy.

  5. Notation The usual notation for "event A occurs given that event B has occurred" is A|B (A given B). The symbol | is a vertical line and does not imply division. P(A|B) denotes the probability that event A will occur given that event B has occurred already.

  6. Suppose we would like to know the probability of selecting a calculus student knowing that the student is taking Data Management. Without new Restrictions: P(C) P(C)= 80/110 But -> condition – must be already enrolled in Data Hence we must modify our Sample Space Only D P(C|D) = n(C∩D) / n(D) = 50/68 17 1 45 5 2 5 25 C Revisit the Original Problem 10

  7. Conditional Probability A rule that can be used to determine a conditional probability from unconditional probabilities is P(A|B) = n(A ∩ B) = P(A andB) n(B) P(B) where, P(A|B) = the (conditional) probability that event A will occur given that event B has occurred already P(A ∩ B) = the (unconditional) probability that event A and event B occur P(B) = the (unconditional) probability that event B occurs

  8. 17 1 45 5 2 5 25 C Find P(G | D) 10

  9. Problem • You roll one die twice. What is the probability of obtaining a sum greater than 8 if you roll a 5 on the first roll

  10. Problem • A bag contains Three White balls numbered 1, 2, and 3, One Red ball numbered 4, and Four Blue balls numbered 5, 6, 7, and 8. A ball is selected at random. What is the probability that the ball chosen is White, given that the chosen ball is marked with an odd number?

  11. Multiplication Law for Conditional Probability • The probability of events A and B occurring, given that A has occurred, is given by P(B ∩ A) = P(B|A) x P(A) Just a rearrangement of the previous formula

  12. Example • What is the probability of drawing 2 face cards in a row from a deck of 52 playing cards if the first card is not replaced? P(1st FC ∩ 2nd FC) = P(2nd FC | 1st FC) x P(1st FC) = 11 x 12 51 52 = 132 2652 = 11 221

  13. Exercises / Homework • Homework: • Read Examples 1-3, pp. 231 – 234 • Complete pp.235-238 # 1-10

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