CISC 1100 Counting and Probability

Counting is Based on Straightforward Rules 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?

Counting Can be Summarized as Follows 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 Picking Cards 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 Picking License Plates 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 =

Probability 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

Probability is just counting (Twice) 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 Horse Racing 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 Horse Racing 2 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 Electing Class Officers 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 different groupings you could be part of

Class Officers (cont) 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 Card Example Revisited 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! = So your probability of getting 4 aces is 48/

Trick about order mattering 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.

Other Ideas 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