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CSRU 1100 Counting and Probability

CSRU 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?

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CSRU 1100 Counting and Probability

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  1. CSRU 1100Counting and Probability

  2. 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?

  3. 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.

  4. 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 5th card. • 4 * 3 * 2 * 1 (but order does not matter so divide by 4!) • 48 choices for the 5th card. • 1 * 48 = 48

  5. ExamplePicking 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 = 1091750400

  6. 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

  7. 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

  8. 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

  9. ExampleHorse 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

  10. ExampleElecting 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. • 1026 different groupings you could be part of

  11. 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

  12. ExampleCard 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! = 2598960 • So your probability of getting 4 aces is 48/2598960

  13. 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.

  14. 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

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