Section 3-3 Mutually exclusive events are events that cannot both happen at the same time. The Addition Rule (For “OR” probabilities) “Or” can mean one of three things: A occurs and B does not occur. B occurs and A does not occur. Both A and B occur. If A and B are mutually exclusive, simply add their probabilities of occurring together. If A and B are NOT mutually exclusive, add their probabilities of occurring together and then subtract the probability that they both occur.

Type of Probability & Rules In Words In Symbols Classical Probability The number of outcomes in the sample space is known and each outcome is equally likely to occur 𝑃 𝐸 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐸 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Empirical Probability The frequency of outcomes in the sample space is estimated from experimentation (you have data) 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝐸 𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑓 𝑛 Range of Probabilities Rule ALL probabilities are between zero and 1, inclusive 0≤𝑃(𝐸)≤1 Complementary Events The complement of event E is the set of all outcomes in a sample space that are NOT included in E, denoted E’ 𝑃 𝐸′ =1−𝑃 𝐸 ; 𝑃 𝐸 =1−𝑃 𝐸′ Multiplication Rule The Multiplication Rule is used to find the probability of two events occurring in a sequence (AND) 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∗𝑃 𝐵 𝐴 (𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡) 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∗𝑃 𝐵 (𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡) Addition Rule The Addition Rule is used to find the probability of at least one of two events occurring (OR) 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 (𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒)

Example 1: Determine whether these two events are mutually Example 1: Determine whether these two events are mutually exclusive or not. 1) Roll 3 on a die AND roll 4 on a die. 2) Randomly select a male student AND randomly select a nursing major. 3) Randomly select a blood donor with type O blood AND randomly select a female blood donor. 4) Randomly select a jack from a deck of cards AND randomly select a face card from a deck of cards. 5) Randomly select a 20-year-old student AND randomly select a student with blue eyes. 6) Randomly select a vehicle that is a Ford AND randomly select a vehicle that is a Toyota.

SOLUTIONS 1) You cannot roll a 3 AND a 4 on the same roll. These are mutually exclusive. 2) You can be both a male and a nursing major. These are not mutually exclusive. 3) You can be both a female and have type 0 blood. These are not mutually exclusive. 4) Jacks are face cards, so any jack is also a face card. These are not mutually exclusive. 5) You can be a 20-year-old student and have blue eyes. These are not mutually exclusive. 6) A vehicle can not be BOTH a Ford AND a Toyota at the same time. These are mutually exclusive.

Example 2: 1) You select a card from a standard deck. Find the probability that the card is a 4 or an ace. 2) You select a card from a standard deck. Find the probability that the card is a queen or a red card. SOLUTIONS 1) “Ace” and “4” are mutually exclusive. Add the probability of getting an ace to the probability of getting a 4. The probability of getting an ace is 4 52 ; the probability of getting a four is also 4 52 . The probability of getting an ace or a four is 4 52 + 4 52 = 8 52 = 2 13 ≈0.154

2) The events are not mutually exclusive (there are 2 red queens). Add the probability of getting a queen to the probability of getting a red card, then subtract the probability of getting a red queen. P(queen) = 4 52 ; P(red) = 26 52 ; P(red queen) = 2 52 4 52 + 26 52 − 2 52 = 30 52 − 2 52 = 28 52 ≈0.538

Example 3 The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month? Sales Volume 0-24,999 25,000-49,999 50,000-74,999 75,000-99,999 100,000-124,999 125,000-149,999 150,000-174,999 175,000-199,999 Months 3 5 6 7 9 2 1

Define Event A as monthly sales between $75,000 and $99,999 SOLUTION: Define Event A as monthly sales between $75,000 and $99,999 Define Event B as monthly sales between $100,000 and $124,999 Because events A and B are mutually exclusive, the probability that the sales representative will sell between $75,000 and $124,999 next month is: 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 7 36 + 9 36 = 16 36 = 4 9 ≈0.444 Sales Volume 0-24,999 25,000-49,999 50,000-74,999 75,000-99,999 100,000-124,999 125,000-149,999 150,000-174,999 175,000-199,999 Months 3 5 6 7 9 2 1

Example 4 A blood bank catalogs the types of blood, including positive or negative Rh-factor, given by donors during the last five days. The number of donors who gave each blood type is shown in the table. A donor is selected at random. 1. Find the probability that the donor has type O or type A blood. 2. Find the probability that the donor has type B or is Rh- negative.   Blood Type O A B AB Total Rh-Factor Positive 156 139 37 12 344 Negative 28 25 8 4 65 184 164 45 16 409

Blood Type O A B AB Total Rh-Factor SOLUTIONS: 1. These events are mutually exclusive 𝑃 𝑂 𝑜𝑟 𝐴 =𝑃 𝑂 +𝑃 𝐴 184 409 + 164 409 = 348 409 ≈0.851 2. These events are not mutually exclusive. 𝑃 𝐵 𝑜𝑟 𝑅ℎ−𝑛𝑒𝑔 =𝑃 𝐵 +𝑃 𝑅ℎ−𝑛𝑒𝑔 −𝑃(𝐵 𝑎𝑛𝑑 𝑅ℎ−𝑛𝑒𝑔) 45 409 + 65 409 − 8 409 = 102 409 ≈0.249   Blood Type O A B AB Total Rh-Factor Positive 156 139 37 12 344 Negative 28 25 8 4 65 184 164 45 16 409

Example 5 Use the graph to find the probability that a randomly selected draft pick is NOT a running back or a wide receiver. SOLUTION: Define Events A and B Event A: Draft pick is a running back. Event B: Draft pick is a wide receiver. These events are mutually exclusive, so the probability that the draft pick is a running back or a wide receiver is: 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 25 255 + 34 255 = 59 255 ≈0.231 To find the probability that the draft pick is NOT a running back or wide receiver, simply subtract the probability that he was one of those positions from 1 (complement rule) 1−0.231=0.769

Assignment: Classwork: Pages 165 #1–12 All Homework: Pages 166 #14–26 Evens