Introductory Statistics, Lecture 9

5.2 Addition Rule and Disjoint Events Math 140 Introductory Statistics „ First, “OR” in mathematics means one, the other, or both. „ Two events A and B are called disjoint (mutually exclusive) if they have no outcomes in common. „ If A and B are disjoint then Lecture 9 P (A or B) = P (A) + P (B) „ Similarly if A, B, and C are mutually exclusive then P (A or B or C) = P (A) + P (B) + P (C) 1 Discussion A or B Type of Fishing All freshwater fishing Saltwater fishing Total 2 Discussion A or B „ Are the categories in the Number table of Display 6.8 (Thousands) complete? Are they disjoint? 31,041 „ What is the probability that a randomly selected person who fishes does their fishing in freshwater or in 8,885 saltwater? How many people fish in both 35,578 freshwater and saltwater? Type of Fishing All freshwater fishing Saltwater fishing Total 3 „ Are the categories in the Number table of Display 6.8 (Thousands) complete? Are they disjoint? 31,041 „ Complete: YES, any person that fishes does so in either fresh water or salt water (maybe both) 8,885 „ Disjoint: NO, the events “Saltwater” and “Freshwater” 35,578 have outcomes in common. 4 Discussion A or B Type of Fishing Number (Thousands) All freshwater 31,041 fishing Saltwater fishing 8,885 Total 35,578 Discussion A or B „ What is the probability that a randomly selected person who fishes does their fishing in freshwater or in saltwater? How many people fish in both freshwater and saltwater? „ P (“Fresh” or “Salt”) = 1 However, 31041 P (“Fresh”) = 35578 P (“Salt”) = Type of Fishing Number (Thousands) All freshwater fishing 31,041 Saltwater fishing 8,885 Total 35,578 Fresh Water 8885 35578 and then, 39926 P (“Fresh”) + P (“Salt”) = 35578 Salt Water „ What is the probability that a randomly selected person who fishes does their fishing in freshwater or in saltwater? How many people fish in both freshwater and saltwater? Similarly # (“Only Fresh”) = 35578 – 8885 # (“Only Fresh”) = 26693. # (“Only Salt” or “Only Fresh”) = 31230 # (“Only Salt”) + #(“Fresh”) = 35578 # (“Only Salt”) + 31041 = 35578 # (“Only Salt”) = 35578 – 31041 = 4537 >1 Then, #(“Fresh” and “Salt”) = 35578 – 31230 #(“Fresh” and “Salt”) = 4348 5 6 A Property of Disjoint Events The addition rule with dice „ If A and B are disjoint then „ Use the Addition Rule for Disjoint Events to find the probability that if you roll two dice, you get a sum of 7 or a sum of 8. P (A and B) = 0 „ Example. Suppose two dice are rolled. A is the event of getting a sum of 12, B is the event of getting two odd numbers. What is P (A and B) ? „ The only way to get a sum of 12 is (6,6) and both numbers are even. So A and B are disjoint and then P (A and B) = 0. 7 8 Discussion D11 General Addition Rule „ Suppose you select a person at random from „ For any two events A and B, your school. Which of these pairs of events must be disjoint? P (A or B) = P (A) + P (B) – P (A and B) a. the person has ridden a roller coaster; the person has ridden a Ferris wheel „ b. owns a classical music CD; owns a jazz CD „ c. is a senior; is a junior „ d. has brown hair; has brown eyes „ e. is left-handed; is right-handed „ f. has shoulder-length hair; is a male „ „ In particular if A and B are disjoint then P (A and B) = 0 and then, P (A or B) = P (A) + P (B) 9 10 Example: The Addition Rule for Events that are not Disjoint (p. 320) Example: Computing P (A and B) „ Use the Addition Rule to find the probability „ In a local school, 80% of the students carry a backpack, B, or a wallet, W. Forty percent carry a backpack and 50% carry a wallet. If a student is selected at random, find the probability that the student carries both a backpack and a wallet. that if you roll two dice, you get doubles or a sum of 8. „ Second Solution. Doubles? Sum of 8? Yes No Total Yes 1 4 5 No 5 26 31 Total 6 30 36 P = 1 + 5 + 4 10 = 36 36 11 12 5.4 Conditional Probability Definition of Conditional Probability „ The Titanic sank in 1912 without enough lifeboats for „ Conditional Probability refers to the the passengers and crew. Almost 1500 people died, most of them men. Was that because a man was less likely than a woman to survive? Or did more men die simply because men outnumbered women by more than 3 to 1 on the Titanic? probability of a particular event where additional information is known. „ We write it the following way P (S |T ) = “probability of S given that T is known to have happened” „ For short we refer to P (S |T ) as the probability of S given T. Gender Survived? Male Female Total Yes 367 344 711 No 1364 126 1490 Total 1731 470 2201 13 Example: The Titanic and Conditional Probability (p. 326) „ Suppose you pick a person at random from the list of people aboard the Titanic. Let S be the event that this person survived, and let F be the event that the person was female. Find P (S |F ) Gender Yes Survived? No Male Female Total 367 344 711 1364 126 1490 Total 1731 470 2201 Discussion P23. (p. 334) „ For the Titanic data, let S be the „ P (F) = Probability of a female passenger event a person survived and F 470 be the event a person was P (F ) = ≈ 0.213 female. Find and interpret these 2201 probabilities. „ a. P (F) and P (F |S ) „ P (F |S ) = Probability that a survivor „ b. P (not F), P (not F |S), and is female. P(S |not F) „ To find P (S |F ) restrict your sample space to only the 470 females (the outcomes for which we know the condition F is true). „ Then among these the favorable outcomes are 344. Thus P (S | F ) = 14 P (F | S ) = Gender 344 ≈ 0.732 472 „ So 73.2 % of the females Survived? survived even though only 32.3 % of the people survived. 15 Male Female Total Yes 367 344 711 No 1364 126 1490 Total 1731 470 2201 344 ≈ 0.483 711 „ P (not F) = Probability of a male passenger P (not F ) = 1731 ≈ 0.786 2201 16 The Multiplication Rule for P (A and B) (p. 327) Discussion P23. (p. 334) „ For the Titanic data, let S be the „ P (not F |S ) = Probability that a survivor is male. event a person survived and F be the event a person was 367 female. Find and interpret these P (not F | S ) = ≈ 0.516 711 probabilities. „ a. P (F) and P (F |S ) „ b. P (not F), P (not F |S), and „ P (S | not F ) = Probability of P(S |not F) surviving given that the person selected is male. Gender Survived? Male Female Total Yes 367 344 711 No 1364 126 1490 Total 1731 470 2201 P (S | not F ) = 367 ≈ 0.212 1731 We can interpret the top branch as: P (F ) ⋅ P (S | F ) = 470 344 344 ⋅ = = P (F and S ) 2201 470 2201 17 18 The Multiplication Rule Definition of Conditional Probability „ The probability that event A and event B both „ For any two events A and B, P (B)>0, happen is given by P (A and B) = P (A).P (B | A) or alternatively P (A and B) = P (B).P (A | B) P (A and B ) P (B ) „ Example: Suppose you roll two dice. Use the definition of conditional probability to find the probability that you get a sum of 8 given that you rolled doubles. P (A | B ) = P (sum 8 and doubles ) P (doubles) 1 / 36 1 = = 6 / 36 6 P (sum 8 | doubles ) = 19 20 Important Uses of Conditional Probability „ To compare sampling with or without replacement. „ To study effectiveness of medical tests „ To study effectiveness of statistical testing 21