658 CHAPTER 11 Probability m A die (one of a pair of dice) contains six surfaces, called faces. Each face contains a unique number of dots, from 1 to 6. The sum of the dots on opposite surfaces is 7. StatCrunch Applets Experiment Flip Coin When a coin is tossed, one example of an event would be getting a head. A different event would be getting a tail. When a single die is rolled, there are many different events such as rolling an even number, rolling a number less than 4, rolling a 5, and rolling a number greater than 2. When a card is drawn from a standard deck of playing cards, there are many different events such as drawing an ace, drawing a face card (jack, queen, or king), drawing a club, or drawing a red card. When a researcher administers an experimental drug, one example of an event is the patient has a favorable reaction to the drug. Once we understand the terms experiment, outcome, and event, we can discuss probability. The probability of an event is a numeric measure of the likelihood that the event will occur. In probability, we often symbolize an event with the letter E. Given an event E, we will symbolize the probability of E by P E( ), which is read “P of E.” Probability is classified as either empirical (experimental) or theoretical (mathematical). Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Determining the chance of something happening in the future by observing past results is called empirical probability. Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment. Empirical Probability and the Law of Large Numbers We will first briefly discuss empirical probability. The emphasis later in this section and in the remaining sections will be on theoretical probability. Following is the formula for computing empirical probability, or relative frequency. Definition: Outcomes The possible results of an experiment are called its outcomes. The outcomes of tossing a coin are heads or tails. The outcomes of rolling a single die are the numbers 1, 2, 3, 4, 5, and 6. The outcomes of drawing a single card from a standard deck of playing cards are the 52 different cards. The possible outcomes from administering an experimental drug may be a favorable reaction, no reaction, or an adverse reaction. Definition: Event An event is a subcollection of the outcomes of an experiment. Empirical Probability (Relative Frequency) The empirical probability of an event, P E( ), can be determined by the following formula. = P E E ( ) number of times event has occurred total number of times the experiment has been performed The probability of an event, whether empirical or theoretical, is always a number between 0 and 1, inclusive, and may be expressed as a decimal number or a fraction. An empirical probability of 0 indicates that the event has never occurred. An empirical probability of 1 indicates that the event has always occurred.
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