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Introduction Probability Introduction; Experiments, Outcomes, & Events Probability of an Event The probability of an event is a number between 0 and 1 that expresses the longrun likelihood that the event will occur. An event having probability .1 is rather unlikely to occur. An event with probability .9 is very likely to occur. An event with probability .5 is just as likely to occur as not. Example: Experiment, Trial and Outcome Experiment 1: Flip a coin Trial: One coin flip Outcome: Heads Experiment 2: Allow a conditioned rat to run a maze containing three possible paths Trial: One run Outcome: Path 1 Experiment 3: Tabulate the amount of rainfall in New York, NY in a year Trial: One year Outcome: 37.23 in In this chapter, we discuss probability, which is the mathematics of chance. Many events in the world around us exhibit a random character, but by repeated observations of such events we can often determine longterm patterns (despite random, short-term fluctuations). Probability is the branch of mathematics devoted to the study of such events. Experiment, Trial, & Outcome An experiment is an activity with an observable outcome. Each repetition of the experiment is called a trial. In each trial we observe the outcome of the experiment. Sample Space The set of all possible outcomes of an experiment is called the sample space of the experiment. So each outcome is an element of the sample space. There are two types of sample spaces: finite and infinite. Note: The sample space of an event is equivalent to the universal set. 1 Example: Sample Space An experiment consists of throwing two dice, one red and one green, and observing the numbers on the uppermost face on each. What is the sample space S of this experiment? Each outcome of the experiment can be regarded as an ordered pair of numbers, the first representing the number on the red die and the second the number on the green die. Example: Sample Space Event S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} Example: Event For the experiment of rolling two dice, describe the events E1 = {The sum of the numbers is greater than 9} An event E is a subset of the sample space. We say that the event occurs when the outcome of the experiment is an element of E. E2 = {The sum of the numbers is 7 or 11}. Events as Sets Special Events Let E and F be two events of the sample space S. Let S be the sample space of an experiment. The event corresponding to the empty set, Ø, is called the impossible event, since it can never occur. The event corresponding to the sample space itself, S, is called the certain event because the outcome must be in S. The event where either E or F or both occurs is designated by E ∪ F. The event where both E and F occurs is designated by E ∩ F. The event where E does not occur is designated by E '. 2 Example: Events As Sets For the experiment of rolling two dice, let E1 = “The sum of the numbers is greater than 9” and E3 = “The numbers on the two dice are equal”. Determine the sets: E1 ∪ E3 Mutually Exclusive Events Let E and F be events in a sample space S. Then E and F are mutually exclusive (or disjoint) if E∩F=Ø If E and F are mutually exclusive, then E and F cannot simultaneously occur; if E occurs, then F does not; and if F occurs, then E does not. E1 ∩ E3 Example Example: Mutually Exclusive Events For the experiment of rolling two dice, which of the following events are mutually exclusive? E1 = “The sum of the dots is greater than 9” E2 = “The sum of the dots is 7 or 11” E3 = “The dots on the two dice are equal” A letter is selected at random from the word “ALABAMA.” a.) What is the sample space for this experiment? b.) Describe the event “the letter chosen is a vowel” as a subset of the sample space. Example Example An experiment consists of tossing a coin four times and observing the sequence of heads and tails. a.) What is the sample space of this experiment? b.) Determine the event E 1 = “more heads than tails occur.” c.) Determine the event E 2 =“the first toss is a tail.” Suppose that you observe the time (in minutes) that it takes a bank teller to deal with a customer. Describe the sample space. Is this a finite or infinite sample space? d.) Determine the event E 1 ∩ E 2. 3 Example: The Game of Clue Example Consider the following events: A = “a person has a dog” B = “a person is taking a math class” C = “ a person does not have any pets” D = “a person is taking an English class” Which, if any, of the events would be mutually exclusive? Explain. Anthony E. Pratt, then inventor of the game Clue, died in 1996. Clue is a board game in which players are given the opportunity to solve a murder that has six suspects, six possible weapons, and nine possible rooms, where the murder may have occurred. The six suspects are Colonel Mustard, Miss Scarlet, Professor Plum, Mrs. White, Mr. Green, and Mrs. Peacock. Example: The Game of Clue (continued) How could a sample space be formed with the entire solution to the murder, giving murderer, weapon, and site? How many outcomes would the sample space have? Let E be the event the murder occurred in the library. Let F be the event that the weapon was a gun. Describe E ∪ F and E ∩ F. 4