Expected value a weighted average of all possible

Expected value a weighted average of all possible values : where the weights are the probabilities of each outcome

Expected Value • If a random variable x can have any of the values x 1, x 2 , x 3 , … • the corresponding probabilities of these values occurring are P(x 1), P(x 2), P(x 3), … • then the expected value of x is given by © 2008 Pearson Addison. Wesley. All rights reserved 12 -5 -2

Example: expected value probability distribution of ER arrivals x is the number of arrivals in one hour X 10 11 12 13 14 P(x) . 4 . 2 . 1 The average expected number each hour

© 2008 Pearson Addison. Wesley. All rights reserved Try on Excel: Expected Value Children in third grade were surveyed and told to pick the number of hours that they play electronic games each day. The probability distribution is given below. # of Hours x Probability P(x) 0 . 3 1 . 4 2 . 2 3 . 1 Compute a “weighted average” by multiplying each possible time value by its probability and then adding the products.

Expected Value Compute a “weighted average” by multiplying each possible time value by its probability and then adding the products. 1. 1 hours is the expected value (or the mathematical expectation) of the quantity of time spent playing electronic games. © 2008 Pearson Addison. Wesley. All rights reserved 12 -5 -5

Fundamental Counting Principle if an event has m possible outcomes and another independent event has n possible outcomes, then there are m* n possible outcomes for the two events together. EXAMPLE A student is to roll a die and flip a coin. How many possible outcomes will there be? 6*2 = 12 outcomes

Fundamental Counting Principle For a college interview, Robert has to choose what to wear from the following: 4 slacks 3 shirts 2 shoes 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits

Permutations dependent events and order matters A Permutation is an arrangement of items in a particular order. To find the number of Permutations of n items chosen r at a time, you can use the formula

Permutations: dependent events The number of ways to arrange the letters ABC: ____ 3 ____ Number of choices for second blank? 3 2 ___ Number of choices for third blank? 3 2 1 Number of choices for first blank? 3*2*1 = 6 ABC ACB BAC 3! = 3*2*1 = 6 BCA CAB CBA

Permutations Examples A combination lock will open when the right choice of three numbers (from 1 to 30) is selected. How many different lock combinations are possible assuming no number is repeated?

Permutation Example From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

Combinations A Combination is an arrangement of items in which order does not matter. There always fewer combinations than permutations. To find the number of Combinations of n items chosen r at a time, you can use the formula

Combinations Example To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

Combinations Practice: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

Combinations Examples A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Center: Forwards: Guards: Thus, the number of ways to select the starting line up is 2*10*6 = 120.