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Probability Math 374

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Game Plan General General Models Models a) Tree Diagram b) Matrix Two Dimensional Model c) Balanced d) Unbalanced Odds – for – odds against Odds – for – odds against

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What is Probability It is a number we assign to show the likelihood of an event occurring It is a number we assign to show the likelihood of an event occurring We set the following limits We set the following limits What is the probability that if I drop the piece of chalk it will fall to the floor? What is the probability that if I drop the piece of chalk it will fall to the floor? P (fall) = 1 a certainly P (fall) = 1 a certainly

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Probability What is the probability that the chalk will float up to the ceiling? What is the probability that the chalk will float up to the ceiling? P (float) = 0 an impossibility P (float) = 0 an impossibility

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Probability Scale We have created a scale We have created a scale 0 Absolute Impossibility 1 Absolute Certainty

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Various Types of Probability Subjective – gets you in trouble Subjective – gets you in trouble Probability – (Canadiens will will Stanley Cup) Probability – (Canadiens will will Stanley Cup) Experimental – you need to do an experiment Experimental – you need to do an experiment Probability (cars on an assembly line have a bad headlight). You would probably test 20 cars. If 1 was faulty you would say 1/20 are bad Probability (cars on an assembly line have a bad headlight). You would probably test 20 cars. If 1 was faulty you would say 1/20 are bad 0 1 0.1 A leafs fan 0.8 (A fan)

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Various Types of Probability Theoretical – the one we will use Theoretical – the one we will use Fundamental Definition Fundamental Definition P = S P = S R where s # of successes R # of possibilities R # of possibilities

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Examples Consider flip a coin, what is the probability of getting a tail Consider flip a coin, what is the probability of getting a tail S = (T) = 1 S = (T) = 1 R = (H, T) = 2 R = (H, T) = 2 P = ½ P = ½

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Examples Roll a die, get a 5 Roll a die, get a 5 S = (5) = 1 S = (5) = 1 R = (1,2,3,4,5,6) = 6 R = (1,2,3,4,5,6) = 6 P = 1/6 P = 1/6 Roll a die, get more than 2 Roll a die, get more than 2 S = (3,4,5,6) S = (3,4,5,6) R = 6 R = 6 P = 4/6 (you do not need to reduce in this chapter!) P = 4/6 (you do not need to reduce in this chapter!)

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Models The key to understanding probability is to have a model that shows you the possibilities The key to understanding probability is to have a model that shows you the possibilities This can get daunting, there are 311 875 200 possible poker hands from a standard deck. This can get daunting, there are 311 875 200 possible poker hands from a standard deck. The easiest model we will use is a tree The easiest model we will use is a tree Tree Model - Flipping two coins Tree Model - Flipping two coins H T T H T H Starting Point

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We need to Determine R In a balanced model just count the number of end branches i.e. 4 to determine denominator In a balanced model just count the number of end branches i.e. 4 to determine denominator OR 2) R = # of possibilities of first. # of possibilities of second. # of possibilities of third. OR 2) R = # of possibilities of first. # of possibilities of second. # of possibilities of third. 2 x 2 = 4 2 x 2 = 4 Using the model Using the model P (getting two tails) P (getting two tails) S How many branches from start to the end satisfy? S How many branches from start to the end satisfy? Lets look at the various types of models Lets look at the various types of models

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Tree Model H T T H T H Starting Point S = ?S = 1 P = ¼ P = ? Notice # of branches will be the denominator Look at the # of successes for numerator

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Matrix Two Dimensional Model Rolling Two Die or Dice Rolling Two Die or Dice Not a tree Not a tree Called a matrix – two dimensional Called a matrix – two dimensional Eg P (getting a total 5) Eg P (getting a total 5) S = 4 S = 4 P = 4/36 P = 4/36 Roll over 3 Roll over 3 Do not include 3 Do not include 3 P = 33/36 P = 33/36 123456 1234567 2345678 3456789 45678910 567891011 6789101112 Die #1 Die #2

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Balanced Model Consider a bag with 2 blue marbles and 3 red marbles. You are going to pick two and replace them. Consider a bag with 2 blue marbles and 3 red marbles. You are going to pick two and replace them. Replace = put them back Replace = put them back What is the prob of getting a blue & red? What is the prob of getting a blue & red?

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Balanced Model B B R R R B B R B B R R R R R Starting Point B B B B B B R R R R R R # of successes? 12 P = 12/25 # of Possibilities? R = 25 P (blue & Red)? R R R Put check marks!

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Unbalanced Model It is not always possible to write out every single branch. Consider the same question; It is not always possible to write out every single branch. Consider the same question; What is the P of getting a blue and a red? What is the P of getting a blue and a red? This time we create an unbalanced model This time we create an unbalanced model Starting Point 2 3 B B R R R B 2 2 3 3 S? (2x3)+ (3x2) R? P=12/25 R=5x5 To find den. ADD branches and MULT each one. (It differs if you have 3 options).

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Unbalanced Model Create a model given a bag with 20 blue, 15 green and 15 red marbles. You are picking three marbles and replacing them. Create a model given a bag with 20 blue, 15 green and 15 red marbles. You are picking three marbles and replacing them. What is the probability of getting three green? What is the probability of getting three green? Draw the model! Draw the model! S = ? S = ? 15 x 15 x 15 15 x 15 x 15 R = ? R = ? 50 x 50 x 50 50 x 50 x 50 P = 3375 / 125000 P = 3375 / 125000

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Unbalanced Model What is the probability of getting a blue, a green and a red? What is the probability of getting a blue, a green and a red? Since they do not mention it, we must assume order does not matter. Since they do not mention it, we must assume order does not matter. We need to look at BGR, BRG, GRB, GBR, RBG and RGB. We need to look at BGR, BRG, GRB, GBR, RBG and RGB. S = (20x15x15) + ? S = (20x15x15) + ? + (20x15x15) + (15x15x20) + (15x20x15) + (15x20x15) + (15x15x20) = 27000 P = 27000 / 125000 P = 27000 / 125000

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Without Replacement Without replacement = not putting them back (you have less possibilities afterwards) Without replacement = not putting them back (you have less possibilities afterwards) Given a bag with 5 red, 10 blue and 15 green and you will pick three marbles and do not replace them. Given a bag with 5 red, 10 blue and 15 green and you will pick three marbles and do not replace them. Create a model Create a model

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Without Replacement What is the probability of getting a B-R-G in any order? (5 red, 10 blue and 15 green) What is the probability of getting a B-R-G in any order? (5 red, 10 blue and 15 green) So we are looking at RBG, RGB, BRG BGR GRB GBR So we are looking at RBG, RGB, BRG BGR GRB GBR S = (5x10x15) + (5x15x10) + (10x5x15) + (10x15x5) + (15x5x10) + (15x10x5) = 4500 S = (5x10x15) + (5x15x10) + (10x5x15) + (10x15x5) + (15x5x10) + (15x10x5) = 4500 R = ? R = ? P = 4500 / 24360 P = 4500 / 24360 R = 30 x 29 x 28 = 24360

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Without Replacement What is the probability of getting 2 B and one G or two G and one B? What is the probability of getting 2 B and one G or two G and one B? So we are looking at BBG BGB GBB GGB GBG BGG So we are looking at BBG BGB GBB GGB GBG BGG S = (10x9x15) + (10x15x9) + (15x10x9) + (15x14x10) + (15x10x14) + (10x15x14) = 10350 S = (10x9x15) + (10x15x9) + (15x10x9) + (15x14x10) + (15x10x14) + (10x15x14) = 10350 P = 10350 / 24360 P = 10350 / 24360 Do Stencil #5,6,7 Do Stencil #5,6,7

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Odds For – Odds Against Another way of showing a situation in probability is by odds Another way of showing a situation in probability is by odds Note: These are not bookie odds – that is subjective probability! Note: These are not bookie odds – that is subjective probability! We have so far P = S We have so far P = S R We will now define F as the number of failures. Thus S + F = R We will now define F as the number of failures. Thus S + F = R # of Successes + # of Failures = # of Possibilities # of Successes + # of Failures = # of Possibilities

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Odds For Odds for are stated S : F Odds for are stated S : F Eg The odds for flipping a coin and getting a head is 1:1 Eg The odds for flipping a coin and getting a head is 1:1 Eg The odds for flipping two coins and getting two heads 1:3 Eg The odds for flipping two coins and getting two heads 1:3

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Odds Against Odds against are stated F : S Odds against are stated F : S Eg The odds against flipping two coins and getting two heads Eg The odds against flipping two coins and getting two heads 3:1 3:1 If the odds for an event are 8:3, what is the probability? If the odds for an event are 8:3, what is the probability? S = 8, F = 3 Thus R = 8 + 3 = 11 S = 8, F = 3 Thus R = 8 + 3 = 11 P = 8 / 11 P = 8 / 11

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Last Question Last Question If the odds against are 9:23, what are the odds for and probability If the odds against are 9:23, what are the odds for and probability 23:9 23:9 P = 23/32 P = 23/32 Do Stencil #8, & #9 Do Stencil #8, & #9

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