Learning Goals: * Define a discrete random variable * Applying a probability distribution of a discrete random variable. * Use tables, graphs, and expressions to represent the distributions. Should you close down a manufacturing plant because 5% of the drills produced during a specific shift were defective? Should you follow a prescribed medical treatment if you are not sure that the test results correctly identified your illness? Probability distributions (theoretical and experimental) and expected value are significant pieces of information that can be used in making these kinds of decisions.
Recall Some probabilities are determined from repeated experimentation and observation, recording results, and then using these results to predict expected probability. This kind of probability is referred to as experimental probability. A frequency distribution is a table that shows the number of data observations that fal into specific intervals. A frequency histogram can be constructed to illustrate the distribution. Random Variable A Random variable is an outcome that takes on a numerical value as a result of an experiment. The value of the random variable, which is not known with certainty before the experiment, is often denoted by X. All random variables are not created equal. A random variable can be classified as either discrete or continuous depending on the numerical values it assumes. A random variable is discrete if it is limited to assuming only specific integer values as a result of counting the outcome of an experiment. A random variable is continuous if it can assume any numerical value within an interval as a result of measuring the outcome of an experiment. Examples : The sum of the dice when a pair of fair dice are rolled (discrete) The number of puppies in a litter (discrete) The return on an investment (continuous) The lifetime of a flashlight battery (continuous)
Discrete Probability Distributions A listing of all the possible outcomes of an experiment for a discrete random variable along with the relative frequency or probability of each outcome is called a discrete probability distribution. This function may be presented as a table of values, a graph, or a mathematical expression. The distributions can involve outcomes with equal or different likelihoods Have applications in many fields including science, game theory, economics, telecommunications, and manufacturing.
Probability Distributions of a Discrete Random Variable Discrete Probability Distributions have 3 major properties: 1. Each outcome in the distribution needs to be mutually exclusive that is, the value of the random variable cannot fall into more than one of the frequency distribution classes. 2. The probability of each outcome P(X), must be between 0 and 1. 0 P(X) 0 for all values of X. 3. P(X) = 1 ( Probabilities must add up to one ) Recall Multiplicative Principle for Counting The total number of possible outcomes in an experiment is found by multiplying the possible outcomes at each step in the sequence. Factorial Notation Allows us to represent, and quickly calculate, the number of different ways that a set of objects can be arranged. The existence of duplicate items changes this slightly.
Flipping Coins Activity Divide into groups of 5 and complete the activity Flipping Coins Step #2 Frequency Step #3 Experimental Probability Mean number of Heads (2.5) # Heads 0 1 2 3 4 5 N(X) 2 3 4 6 4 1 P(x) = N (X)/20 0.1 0.15 0.2 0.3 0.2 0.05
Step #4 Theoretical Probability Results Probability (fraction) Probability (decimal) 5T 1 1/32 0.03125 1H + 4T 5 5/32 0.15625 2H + 3T 10 10/32 0.3125 3H + 2T 10 10/32 0.3125 4H + 1T 5 5/32 0.15625 5H 1 1/32 0.03125 a b c
In groups you will present your answers to the class for the following questions: 1. The manager of a telemarketing firm conducted a time study to analyze the length of time his employees spent engaged in a typical sales-related phone call. The results are shown in the table below, where time has been rounded to the nearest minute. a) Define the random variable X. b) Create a probability distribution for these data. c) Determine the expected length of a typical sales-related call. Time (min) 1 2 3 4 5 6 7 8 9 10 Frequency 15 12 18 22 13 10 5 2 2 1 2. A drawer contains four red socks and two blue socks. Three socks are drawn from the drawer without replacement. a) Create a probability distribution in which the random variable represents the number of red socks. b) Determine the expected number of red socks if three are drawn from the drawer without replacement. 3. Find a set of data that meets the following two conditions: * the data are represented by a frequency table. * a discrete random variable can be used to represent the outcomes. a) Create a probability distribution for your data set. b) Use your data set to determine the expected value. 4. The graph below shows the probabilities of a variable, N, for the values of N = 0 to N = 4. Is this the graph of a valid probability distribution? Explain. P(N = n) n