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

2. Expected Value
If a random variable x can have any of the values x1, x2 , x3 ,…
the corresponding probabilities of these values occurring are P(x1), P(x2), P(x3), …
then the expected value of x is given by
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3. 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

4. Try on Excel: Expected Value
© 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) .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.

5. 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

6. 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

7. 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

8. dependent events and order matters
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

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

10. 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?

11. 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?

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

13. 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?

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

15. 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.