MATH 1680 Chapter 5 Notes

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Collin County Community College District **We aren't endorsed by this school
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MATH 1342
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Statistics
Date
Aug 29, 2023
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28
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Chapter 5 Probability 5.1 Probability Terminology and Notation Basic Definitions When discussing probability, an experiment is any process with uncertain results that can be repeated many times. Common probability experiments include: Tossing a coin Rolling a die Drawing a card from a deck Randomly selecting a sample from a population When an experiment is repeated multiple times, each repetition is called a trial A possible result of the experiment is called an outcome The sample space is the set of all possible outcomes. Tossing a Coin: Heads, Tails Rolling a Die: 1, 2, 3, 4, 5, 6 Drawing a Card: A ♠ , 2 ♠ , 3 ♠, 4 ♠ , 5 ♠ , 6 ♠ , 7 ♠, 8 ♠ , 9 ♠ , 10 ♠ ,J ♠,Q♠ ,K ♠ , A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ,J ,Q ,K , A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ,J ,Q ,K , A ♣ , 2 ♣ , 3 ♣, 4 ♣ , 5 ♣ , 6 ♣ , 7 ♣, 8 ♣ , 9 ♣ , 10 ♣ ,J ♣,Q♣ ,K ♣ A collection of outcomes is called an event Events can be described in words or by listing their outcomes. Example. Suppose a standard die is rolled. List all the possible ways of rolling an odd number. 1, 3, 5 Example. Suppose a coin is tossed three times. List all the possible ways of getting at least two heads. H H H H T H H H T T H H 1
Example. Suppose two dice are rolled. List all the possible ways for the sum to be 8. 6+2, 5+3, 4+4, 3+5, 2+6 Example. Suppose two dice are rolled. List all the possible ways of rolling doubles. 1+1, 2+2, 3+3, 4+4, 5+5, 6+6 Is it possible to roll doubles and for the sum to be 8? 4+4 Is it possible to roll doubles and for the sum to be 7? No Two events are mutually exclusive (or disjoint ) if they have no outcomes in common. Hence it is impossible for both events to occur on the same trial of the experiment. Example. A card is drawn from a standard deck. Which of the following events are mutually exclusive? Drawing an eight Drawing a heart Drawing a spade Drawing a face card Drawing an eight and a face card = mutually exclusive Drawing a heart and spade = mutually exclusive 2
If A and B are mutually exclusive events, knowing that A has occurred makes it less likely that B has also occurred. (In fact, it makes it impossible for B to occur.) Similarly, knowing that B has occurred makes it less likely for A to occur. Two events are dependent if knowing that one of the events has occurred makes it more or less likely that the other event has occurred. Two events are independent if neither event affects the likelihood of the other. Example. A coin is tossed twice. Event A is getting heads on the first toss. Event B is getting heads on the second toss. Are A and B independent? Independent Example. Two dice are rolled. Event A is the sum being greater than 10. Event B is rolling a six on the first die. Are these events independent? No, they're dependent Example. A student is selected at random. Event A occurs if the student is taller than 72 inches. Event B occurs if the student's shoe size is greater than 10. Are these events dependent? Yes Example. A student is selected at random. Event A occurs if the student got more than 7 hours of sleep the night before. Event B occurs if the student's shoe size is greater than 10. Are these events dependent? No, they're independent 3
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