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Expected Value Reprise CP Canoe Club Duck Derby Maximum of 2 000 tickets sold $5/ticket Prizes 1) 12 VIP tickets to Cirque du Soleil ($2,000) 2) 32gb BlackBerry Playbook ($600) 3) $250 Should Mr. Lieff buy a ticket?

Expected Value Reprise

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Expected Value Reprise. CP Canoe Club Duck Derby Maximum of 2 000 tickets sold $5/ticket Prizes 1) 12 VIP tickets to Cirque du Soleil ($2,000) 2) 32gb BlackBerry Playbook ($600) 3) $250 Should Mr. Lieff buy a ticket?. Expected Value Calculation. For 2 000 tickets - PowerPoint PPT Presentation

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Dealing With Uncertainty

Expected Value RepriseCP Canoe Club Duck DerbyMaximum of 2 000 tickets sold$5/ticketPrizes1) 12 VIP tickets to Cirque du Soleil ($2,000)2) 32gb BlackBerry Playbook ($600)3) $250Should Mr. Lieff buy a ticket?Expected Value CalculationFor 2 000 ticketsE(X) =1/2000 (2000) + 1/2000(600) + 1/2000(250) = 1.425

Suppose Mr. Lieff receives a hot tip that only 1 500 tickets will be sold.E(X) =1/1500 (2000) + 1/1500(600)+1/1500(250) = 1.9Probability Distributions and Expected ValueChapter 5.1 Probability Distributions and PredictionsMSIP / Home Learning: p. 277 #1-5, 9, 12, 13Probability Distributions of a Discrete Random Variablea discrete random variable X is a variable that can take on only a finite set of valuesfor example, rolling a die can only produce numbers in the set {1,2,3,4,5,6}rolling 2 dice can only produce numbers in the set {2,3,4,5,6,7,8,9,10,11,12}choosing a card from a standard deck (ignoring suit) can produce only the cards in the set {A,2,3,4,5,6,7,8,9,10,J,Q,K}

Probability Distributionthe probability distribution of a random variable x, is a function which provides the probability of each possible value of xmay be represented as a table of values or a graphex, rolling a die:

Probability Distribution for 2 Dice

What would a probability distribution graph for three dice look like?We will try it! Using three dice, figure out how many outcomes there areThen find out how many possible ways there are to create each of the possible outcomesFill in a table like the one belowNow you can make the graphOutcome3456789# ways1Probability Distribution for 3 DiceOutcome345678910# cases13610152128So what does an experimental distribution look like?A simulated dice throw was done a million times using a computer program and generated the following dataWhat is the most common outcome?Does this make sense?

Back to 2 DiceWhat is the expected value of throwing 2 dice?How could this be calculated?So the expected value of a discrete variable X is the sum of the values of X multiplied by their probabilities

Example 1a: tossing 3 coinsWhat is the likelihood of at least 2 heads?It must be the total probability of tossing 2 heads and tossing 3 headsP(X = 2) + P(X = 3) = + = so the probability is 0.5X0 heads1 head2 heads3 headsP(X)Example 1b: tossing 3 coinsWhat is the expected number of heads?It must be the sums of the values of x multiplied by the probabilities of x0P(X = 0) + 1P(X = 1) + 2P(X = 2) + 3P(X = 3)= 0() + 1() + 2() + 3() = 1So the expected number of heads is 1.5X0 heads1 head2 heads3 headsP(X)CombinationsRecall that C(n, r) is the number of ways r objects can be chosen from n when order doesnt matter.

Example 2a: Selecting a Committee of three people from a group of 4 men and 3 womenWhat is the probability of having at least one woman on the team?There are C(7,3) or 35 possible teamsC(4,3) = 4 have no womenC(4,2) x C(3,1) = 6 x 3 = 18 have one womanC(4,1) x C(3,2) = 4 x 3 = 12 have 2 womenC(3,3) = 1 has 3 womenExample 2a contd: selecting a committeeWhat is the likelihood of at least one woman?It must be the total probability of all the cases with at least one womanP(X = 1) + P(X = 2) + P(X = 3) = 18/35 + 12/35 + 1/35 = 31/35X0 women1 woman2 women3 womenP(X)4/3518/3512/351/35Example 2b: selecting a committeeWhat is the expected number of women?0P(X = 0) + 1P(X = 1) + 2P(X = 2) + 3P(X = 3) = 0(4/35) + 1(18/35) + 2(12/35) + 3(1/35)= 1.3 (approximately)X0 women1 woman2 women3 womenP(X)4/3518/3512/351/35MSIP / Home Learningp. 277 #1-5, 9, 12, 13

Pascals TriangleChapter 5.2 Probability Distributions and PredictionsDue now: p. 277 #1-5, 9, 12, 13MSIP/Home Learning: p. 289 #1, 2aceg, 6-8, 11-12

How many routes are there to the top right-hand corner?you need to move up 4 spaces and over 5 spacesThis is the same as rearranging the letters NNNNEEEEEThis can be calculated by C(9,4) or C(9,5)= 126 waysPermutation AND Combination?As a permutation:There are 9 moves 9!However, 4 are identical N moves and 5 are identical E movesDivide by 4! due to identical arrangements of NsDivide by 5! due to identical arrangements of Es 9! 4!5! = 126As a combination:There are 9 moves required _ _ _ _ _ _ _ _ _Choose 4 to be Ns, the rest are Es C(9, 4) = 126Choose 5 to be Es, the rest are Ns C(9, 4) = 12611 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1Pascals Trianglethe outer values are always 1the inner values are determined by adding two values diagonally abovePascals Triangle11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 1sum of each row is a power of 21 = 202 = 214 = 228 = 2316 = 2432 = 2564 = 26Pascals Triangle11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 1Uses?binomial theoremcombinations!e.g. choose 2 items from 5go to the 5th row, the 2nd number = 10 (always start counting at 0)modeling the electrons in each shell of an atom (google Pascals Triangle electron)Pascals Triangle Cool Stuff11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 1each diagonal is summed up in the next value below and to the leftcalled the hockey stick property

there may even be music hidden in ithttp://www.geocities.com/Vienna/9349/pascal.mid

Music: Three Parts in D Alternating OctatonicThe piece of music has three parts:1) A treble part where the center numbers 1, 2 and 6 are the tonic D of the D alternating octatonic scale. The other vertical lines of numbers represent one scale step; a descending scale step to the left and an ascending scale step to the right. All ways of reaching all the numbers in the triangle are mapped to consecutive sixteenths, starting from the top and going down south west first to every new row, inwards. This part is repeated.2) The main treble part. It has the structure A1 B A2 B. It starts with the left side of the symmetric triangle, including the center (A1), the numbers are the number of scale steps to ascend (south east arrows) or descend (south west arrows). After an intermission (B) where this part doubles the first part, the right side is initiated (A2) and it is the mirror image of A1.3) A walking bass part (quarter notes), same as the first treble part but consecutive equal tones are tied. This part is repeated.Playing time: 4' 47".

Pascals Triangle Cool Stuffnumbers divisible by 5similar patterns exist for other numbershttp://www.shodor.org/interactivate/activities/pascal1/

Pascals Triangle can also be seen in terms of combinationsn = 0n = 1n = 2n = 3n = 4n = 5n = 6

Pascals Triangle - Summarysymmetrical down the middleoutside number is always 1second diagonal values match the row numberssum of each row is a power of 2sum of nth row is 2nBegin count at 0number inside a row is the sum of the two numbers above it

And one more thingremember that for the inner numbers in the triangle, any number is the sum of the two numbers above itfor example 4 + 6 = 10this suggests the following:

which is an example of Pascals Identity

For Example

How can this help us solve our original problem? 1 5 15 35 70126 1 4 10 20 35 56 1 3 6 10 15 21 1 2 3 4 5 6 1 1 1 11so by overlaying Pascals Triangle over the grid we can see that there are 126 ways to move from one corner to anotherHow many routes pass through the green square? to get to the green square, there are C(4,2) ways (6 ways)to get to the end from the green square there are C(5,3) ways (10 ways)in total there are 60 waysHow many routes do not pass through the green square? there are 60 ways that pass through the green squarethere are C(9,5) or 126 ways in total then there must be 126 60 = 66 paths that do not pass through the green squareMSIP / Home LearningRead the examples on pp. 281-287The example starting on the bottom of page 287 is importantpage 289 #1, 2aceg, 6-8, 11-12

Pascals TriangleThe Seeds of AutumnOvercast2007Electronica Dance Trance288072.53eng - 0eng - 00001476 0000145E 0000E0E2 0000E7A7 000236F0 00030A29 00008AA4 00009193 0003188D 00015FA1eng - 00000000 00000210 000006F0 0000000000C1CC80 00000000 0057D7C1 00000000 00000000 00000000 00000000 00000000 00000000