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Sec. 3.1 β Basic Concepts of Probability and Counting A _____________ ______________ is an action, or trial, through which specific results are obtained. The result of a single trial in a probability experiment is an ___________. The set of all possible outcomes of a probability experiment is the __________ ___________. An __________ is a subset of the sample space. It may consist of one or more outcomes. A probability experiment consists of tossing a coin, tossing a different coin, then tossing another different coin. Identify the sample space of the experiment. A probability experiment consists of rolling a six-side die, then tossing a coin. Identify the sample space of the experiment. An event that consists of a single outcome is called a _________ _________. Identify whether an event is a simple event from rolling a six-sided die, then tossing a coin. Getting a 6 on the die Getting a 1 and then getting heads on the coin Getting an odd number and then getting tails on the coin Getting a head and an even prime number on the die The Fundamental Counting Principal If one event can occur π ways and a second event can occur π ways, then the number of ways the two events can occur in sequence is π β π ways. Senior Banquet Main Course Side Dessert A license plate consists of 4 digits and one letter. Only odd numbers can be used for the first digit. How many different license plates are there? An access code has 5 digits. The first digit has to be 5 or less and cannot be 0. The last digit cannot be 3. How many different possible access codes are there? Types of Probability __________ (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. π(πΈ) = ππ’ππππ ππ ππ’π‘πππππ ππ ππ£πππ‘ πΈ ππ’ππππ ππ ππ’π‘πππππ ππ π πππππ π ππππ You roll two six-sided dice. Find the probability of each event. Event A: Rolling a 3 on both dice. Event B: Rolling an even number on both dice. Event C: The sum of both numbers is seven. Event D: Both numbers rolled are the same. Event E: The sum of both numbers is prime. ____________ probability is based on observations obtained from probability experiments. The empirical probability of an event E is the relative frequency of event E. π(πΈ) = πΉππππ’ππππ¦ ππ ππ£πππ‘ πΈ π = πππ‘ππ πππππ’ππππ¦ π A company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. The results are displayed in the table below. Response It is a serious problem. It is a moderate problem. It is not a problem. Number of Times, π 123 115 82 What is the probability that a randomly selected person from the community thinks traffic congestion is a serious problem? What is the probability that a randomly selected person from the community thinks traffic congestion is not a problem? Law of Large Numbers As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event. Describe how the law large numbers relates to tossing a coin 10,000 times opposed to just 10 times. Range of Probabilities Rule The probability of an event E is between 0 and 1 inclusive. 0 β€ π(πΈ) β€ 1 The probability of a certain event is ___________. The probability of an impossible event is ___________. An event that occurs with a probability less than ____________ is typically considered unusual and is highly unlikely to occur. Complementary Events The ________________ of event E is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by ________ (βE primeβ). π(πΈ) + π(πΈ β² ) = 1 Consider the probability experiment of rolling a six-sided die then tossing a coin. Find the probability of the following events. Event A: Getting a four and then tails on the coin Event B: Not getting an even number and then getting heads The complement of event A The complement of event B Answer the following questions about the frequency distribution of employee ages at a company. Employee Ages Frequency, π 15 to 24 54 25 to 34 366 35 to 44 233 45 to 54 180 55 to 64 125 65 and over 42 What is the probability a randomly selected employee is between 35 and 44 years old? What is the probability a randomly selected employee is between 25 and 54 years old? What is the probability a randomly selected employee is not over 54 years old? What is the probability a randomly selected employee is not between 15 and 24 years old?