unit 3 seminar: probability and counting techniques
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I have three shirts: white, blue, pink And two skirts: black, tan How many different outfits can I make?TRANSCRIPT
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Unit 3 Seminar: Probability and Counting Techniques
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Counting(Combinatorics)
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I have three shirts: white, blue, pinkAnd two skirts: black, tan
How many different outfits can I make?
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Choose shirt
white blue pinkChoose skirt black tan black tan black tan
3*2 = 6 outfits
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If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices you can make is M*N.
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A person can be classified by eye color (brown, blue, green), hair color (black, brown, blonde, red) and gender (male, female). How many different classifications are possible?
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An ID number consists of a letter followed by 4 digits, the last of which must be 0 or 1. How many different ID numbers are possible?
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A permutation is an ordered arrangement of things. For example, the permutations of the word BAD are:
BAD ABD DABBDA ADB DBA
Note: AAA is not a permutation of BAD
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We can use the counting principle to count permutations.
Example: How many ways can we arrange the letters GUITAR ?
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n! = n(n-1)(n-2) … 1
6! = 6*5*4*3*2*1 = 720
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What about repeats?
Example: How many ways can we arrange the letters MISSISSIPPI ?
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Sometimes we don’t use all of the available items.
Example: How many ways can we arrange three of the letters WINDY ?
“permutations of size 3, taken from 5 things”
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How many ways can a President, Vice President and Secretary be chosen from a group of 10 people?
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How many selections of 2 letters from the letters WIND can be made (order doesn’t matter) ?
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The number of combinations of n things taken r at a time:
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How many ways can three people be chosen from a group of 10 people?
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Basic Probability
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1.) Classical – based on theoryex: games of chance
2.) Empirical – based on historical observations ex: sports betting
3.) Subjective – based on an educated guess or a rational belief in the truth or falsity of propositionssee: “A Treatise on Probability” by John Maynard Keynes
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EXPERIMENT: Throw a single die.
Sample Space S = {1,2,3,4,5,6}An event is a subset of the sample space
Ex: throw an even number E = {2,4,6}
The probability of an event
P(E) = n(E)/n(S) = 3/6 = 1/2
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Select a card from a deck of 52 cards.What is the probability that it is:
1.) an ace2.) the jack of clubs3.) not a queen4.) the king of stars5.) a heart, diamond, club or spade
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A dartboard has the shape shown.
What is P(7) ?
23
4
7
1 5 6
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Prof. Smith’s grades for a course in College Algebra over three years are:
A = 40B = 180C = 250D = 90F = 60
If Jane takes his course, what is the probability that she will get a C or better?
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Odds in favor of an event = P(success) / P(failure)
= P(it happens) / P(it doesn’t happen)
Ex. A coin is weighted so that P(heads) = 2/3. What are the odds of getting heads?
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What are the odds of rolling a 4 with a fair die?
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The probability of rain today is .35. What are the odds in favor of rain today?
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Expected Value
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The average result that would be obtained if an experiment were repeated many times.
Suppose you have as possible outcomes of the experiment events A1 , A2 , A3 with probabilities P1 , P2 , P3
Expected Value = P1* A1 + P2 *A2 + P3 * A3
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An investment club is considering buying a certain stock. Research shows that there is a 60% chance of making $10,000, a 10% chance of breaking even, and a 30% chance of losing $7200.
Determine the expected value of this purchase.
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Game: Blindfolded, throw a dart. What is the expectation?
$5
$1 $10 $20
$50