conditional probability. suppose you roll two dice does the result of one of the dice affect what...

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Conditional Probability

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Page 1: Conditional Probability. 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

Conditional Probability

Page 2: Conditional Probability. 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

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

Page 3: Conditional Probability. 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

Suppose you had a bag of marbles…

Page 4: Conditional Probability. 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

Conditional Probability Formula• The probability that event B occurs given that event

A has occurred is:P(B | A) = P(A ∩ B)

P(A)

Page 5: Conditional Probability. 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

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?

Page 6: Conditional Probability. 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

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?

Page 7: Conditional Probability. 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

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)

Page 8: Conditional Probability. 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

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(1st FC ∩ 2nd FC) = P(2nd FC | 1st FC) x P(1st FC)

Page 9: Conditional Probability. 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

Example 3

• What is the probability that a student takes Mathematics given that he or she takes English?

Course Taken No. of students

English 80

Mathematics 33

French 68

English and Mathematics

30

French and Mathematics

6

English and French

50

All three courses

5

Page 10: Conditional Probability. 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

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

Page 11: Conditional Probability. 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

Homework

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

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