1. Conditional Probability

2. Suppose you roll two dice Does the result of one of the dice affect what the result of the second one will be? No These are independent events Are there some events that affect other events? Yes These are dependent events for dependent events, knowing that one event has occurred affects the probability that another event will occur

3. Suppose you had a bag of marbles…

4. Conditional Probability Formula The probability that event B occurs given that event A has occurred is: P(B | A) =P(A ∩ B) P(A)

5. Example 1 Light 1 and light 2 are together green 60% of the time Light 1 is green 80% of the time What is the probability that light 2 is green, given that light 1 is green?

6. Example 2 The probability that it snows Saturday and Sunday is 0.2 The probability that it snows Saturday is 0.8 What is the probability that it snows Sunday given that it snowed Saturday?

7. Multiplication Law for Conditional Probability The probability of events A and B both occurring, when B is conditional on A is: P(A ∩ B) = P(B|A) x P(A)

8. Example 2 a) 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 (drawing the first face card) = 12/52 Because you do not put that card back into the deck, there are now 51 cards in the deck, and only 11 face cards left Therefore, P(drawing the second face card) = 11/52 P(A ∩ B) = P(B | A) x P(A) P(1 st FC ∩ 2 nd FC) = P(2 nd FC | 1 st FC) x P(1 st FC)

9. Example 3 What is the probability that a student takes Mathematics given that he or she takes English? Course TakenNo. of students English 80 Mathematics 33 French 68 English and Mathematics 30 French and Mathematics 6 English and French 50 All three courses 5

10. Example 3 To answer the question, we need to find: P(Math | English). We know... P(Math | English) = P(Math ∩ English) P(English) Therefore… P(Math | English) = 0.3 = 3 or 0.375 0.8 8

11. Homework Read Examples 1-3, pp. 231 – 234 pp. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19