Probability Math 374 Game Plan n n a

Probability Math 374

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

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

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

Probability Scale n We have created a scale 0 1 Absolute Impossibilit y Absolute Certainty

Various Types of Probability Subjective – gets you in trouble n Probability – (Canadiens will Stanley Cup) n 0 1 0. 1 A leafs fan n n 0. 8 (A fan) 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

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

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

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

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

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

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

Rolling Two Die or Matrix Two Dimensional Dice n Not a tree n Called a matrix – two dimensional 1 2 3 n Eg P (getting a total 5) 1 2 3 4 n S=4 2 3 4 5 n P = 4/36 3 4 5 6 n Roll over 3 4 5 6 7 n Do not include 3 5 6 7 8 n P = 33/36 6 7 8 9 n Die #2 Model Die #1 4 5 6 7 8 9 10 11 12

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

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

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

Unbalanced Model n n n n 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? Draw the model! S=? 15 x 15 R=? 50 x 50 P = 3375 / 125000

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

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

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

Without Replacement n What is the probability of getting 2 B and one G or two G and one B? n So we are looking at BBG BGB GBB GGB GBG BGG n S = (10 x 9 x 15) + (10 x 15 x 9) + (15 x 10 x 9) + (15 x 14 x 10) + (15 x 10 x 14) + (10 x 15 x 14) = 10350 n P = 10350 / 24360 n Do Stencil #5, 6, 7

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

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

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

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