CSRU 1100

• Are countable items combined using the terms such as AND or OR ?
• Are countable items orderable and if so does the order matter in the particular case?
• Do items get reused when you count, or does the use of one item decrease the number of possibilities of the next item?

• Rule #1 : If you can count them on your own, then count them.
• Rule #2 : If terms combine with “ OR ” then you add the numbers.
• Rule #3 : If terms combine with “ AND ” then you multiply the numbers.
• Rule #4 : If the order you select the numbers does not matter (but there is a scenario where they could matter) then divide your answer by n! where n is the numbers of items you are selecting.

Example
• If you have 52 cards in a deck. How many different ways could someone be dealt a 5 card hand that contains 4 Aces.
• You are selecting 5 cards and the order does not matter.
– You are going to be dealt 4 Aces and you are going to be dealt a 5 th card.
– 4 * 3 * 2 * 1 (but order does not matter so divide by 4!)
– 48 choices for the 5 th card.
– 1 * 48 = 48

Example
• Some states have license plates formed with two letters (which must be different) followed by 4 letters or numbers (which can be the same. How many license plates possibilities are there.
• Pick two different letters AND pick 4 letters/numbers.
• Order matters in both cases.
• 26*25 * 36*36*36*36 = 1091750400

• Knowing how to count also gives you the ability to compute the probability of some event.
• General rules about probability
– All probabilities are numbers between 0 and 1
– A probability of 1 means something is absolutely going to happen
– A probability of 0 means something is NOT going to happen

• Each probability is two counting problems.
– Determine how many possibilities you are interested in having occur (this is called the set of outcomes ).
– Determine how many total possibilities of some general event (this is called the sample space )
– Divide the first number by the second – this is your probability

Example
• 21 horses are in the Kentucky Derby. What is the probability of you picking the winner?
– There is only 1 outcome that interests you (the horse you picked winning)
– There are 21 total possible outcomes (each horse could potentially win)
– Probability is 1/21

Example
• What is the probability that you can pick the top three finishers in order?
– Well again, there is only 1 order that interests you.
– There are 21*20*19 different possibilities for the top three to finish (since order matters).
– 1/7980

Example
• If I am going to select 3 people at random from a class of 20 to be president, vice-president and secretary. What is the probability that you are one of the three students.
– How many groups of 3 are you part of?
• There are 19*18*1 ways you could be secretary
• There are 19*1*18 ways you could be VP
• There are 1*19*18 ways you could be president
• You could be President OR VP OR Secretary.
• 1026 different groupings you could be part of

• How many total groups of 3 are there (order matters)
– 20*19*18 = 6840
• Probability that you are in one of the groups is
• 1026/6840 = .15

Example
• What is the probability of being dealt a 5-card hand that contains 4 aces.
• We know from earlier that there are 48 different hands with 4 aces.
• How many different 5 card hands are there (order does not matter)
52 * 51 * 50 * 49 * 48 / 5! = 2598960
• So your probability of getting 4 aces is 48/2598960

• When doing probabilities the order mattering question ultimately goes away.
• As long as you are consistent between what you do with the outcome space and the sample space it won’t matter if you make the wrong decision about order mattering.
• In other words as long as you do the same thing for both the outcome space and the sample space then the ordering info cancels itself out.

• When you look at each possible outcome of an event and determine its probability you will discover that all of the probabilities always add up to 1.
• What are the outcomes of flipping a coin
– Heads – probability ½
– Tails – probability ½
– They add up to 1