4.4 Probability Distributions and Expected Value 4.4P Robability Distributions and Expected Value

4.4 Probability Distributions and Expected Value 4.4P robability Distributions and Expected Value

Objectives: In this section, you will learn the simplex method to solve a linear programming problem. Upon completion you will be able to: • Express the outcomes of an experiment as values of a random variable. • Compute probability distributions of random variables. • Construct histograms to represent probability distributions. • Use histograms to calculate the probabilities of events. • Compute the expected value of a random variable. • Solve real-world applications using expected values. • Explain whether or not a mathematical "game" is fair.

Defining a Random Variable

Deﬁnition

• In probability, a variable is represented by a letter and it represents a quantitative (numerical) value that is measured or observed in an experiment. • A random variable,X, is a rule which assigns a numerical value to each outcome of an experiment. • A probability distribution is used to organize the values of a random variable and their corresponding probabilities.

Example1 For the experiment of rolling to standard six-sided die and noting the sum, let X be the random variable summing the shown values of both die. a. List all values for the random variable, X.

b. Construct the probability distribution for X.

93 © TAMU Deﬁnition You can give a probability distribution as a table or as a graph. The graph of a probability distribution is a

histogram.

Constructing a Histogram A histogram is drawn in the ﬁrst and second quadrants of a coordinate graph. • The x-axis is a number line representing the values of the random variable, X. • The y -axis represents probability, and should be labeled 0 to 1 for the discussions in this class. Note: If probability was given as percentages, the y-axis would be labeled 0 to 100. • A rectangle is constructed at each random variable value with the height corresponding to the probability of the random variable value, and the width being one unit. The value of the random variable should be placed at the center of the width. • All axes must be labeled.

Example2 For the experiment of rolling to standard six-sided die and noting the sum, let X be the random variable summing the shown values of both die. Use the probability distribution from the previous example to create the corresponding histogram.

Example3 Use the previous histogram to compute: a. What is the probability the sum of the two die is between 4 and 9, inclusively?

b. What is the probability the sum of the two die is great than 6?

c. What is the probability the sum of the two die is 13?

Computing Expected Value

Deﬁnition Suppose the random variable, X, can take on the n values x1, x2, x3,..., xn, with the probability that each of these values occurs being p1, p2, p3,..., pn, respectively.

X x1 x2 x3 ··· xn P(X) p1 p2 p3 ··· pn

Then, the expected value of the random variable is

E(X) = x1 p1 + x2 p2 + ... + xn pn.

N Expected value is the average gain or loss of an event if the procedure is repeated many times.

95 © TAMU Example4 What is the expected value for the probability distribution below?

X 2 4 6 8 10 1 3 2 1 1 P(X) 8 8 8 8 8

Deﬁnition

Since the premium is what a person pays to obtain an insurance policy.

Example5 Assume that you purchased a used car for $12,000 and wish to insure it for full replacement value if it is stolen. If there is a 6% chance that the car will be stolen, what should the premium be for this insurance policy, if the insurance company will not make a proﬁt?

Determining if a Mathematical Game is Fair

Deﬁnition

A mathematical game (money is paid to play) is fair if the expected proﬁt for both sides is 0.

In a fair game, the player should pay an amount equal to their expected winnings.

Example6 Suppose you play a game at a local fair that consisting of drawing a single card from a well-shuﬄed standard deck of cards. The game costs $2, and if you draw a face card you win $1, if you draw an Ace you win $10, but if you draw a number card 2 - 10 you lose. What are your expected winnings? Is this a fair game?

Reﬂection:

• Can you quantify the outcomes of an experiment and display their probabilities using a table? • Can you draw a graphical representation of a probability distribution table? • How would you ﬁnd the expected value of a random variable? • How would you ﬁnd the expected value of a quantity in real-world applications, including mathematical games?