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Random Random WalksWalks
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Who’s This Man?
• Burton Gordon Malkiel•American economist and writer
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What is the idea behind random walk theory?
* Random walk also called drunkard’s walk is based on the premise that one takes a number of successive steps.
* Each step is of equal length.
* Each step is independent from another.
* Each step has two paths of equal likelihood.
* The path taken for each step is completely random.
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11 44 66 44 11
4 Heads4 Heads
3 Heads3 Heads 2 Heads2 Heads
1 Heads1 Heads
0 Heads0 Heads
4 Heads4 Heads 3 Heads3 Heads 2 Heads2 Heads 1 Heads1 Heads 0 Heads0 Heads
Heads or TailsHeads or Tails
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1st Coin1st Coin 2nd Coin2nd Coin 3rd Coin3rd Coin 4th Coin4th Coin
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11 1144 66 44
HHHHHHHHHHHTHHHTHHTHHHTHHHTTHHTTHTHHHTHH
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TTTHTTTHTTTTTTTT
Each coin Each coin shows either shows either
Heads or TailsHeads or Tails
44 0033 22 11Number of HeadsNumber of Heads
A tree diagram of A tree diagram of possible outcomespossible outcomes
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22
66 66
1111 11
33 331111
111111
11 55
77 2121151511
353511
1155 11
2121 77111515
113535
66
1010 10104444
2020
11
The distribution of possible The distribution of possible outcomes corresponds to outcomes corresponds to entries in Pascal’s triangleentries in Pascal’s triangle
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Heads or TailsHeads or Tails
LLLLLLLL RLLLRLLLLRLLLRLLLLRLLLRLLLLRLLLR
RRRLRRRLRRLRRRLRRLRRRLRRLRRRLRRR
RRRRRRRRRRLLRRLLRLRLRLRLRLLRRLLRLRRLLRRLLRLRLRLRLLRRLLRR
Left or RightLeft or Right
Binomial ExperimentBinomial ExperimentRepeated identical and independent Repeated identical and independent steps each having exactly two choicessteps each having exactly two choices
11 1144 66 44 11 1144 66 44
11 1166 4444
11 1144 66 44 11 1144 66 44
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A Binomial Random WalkA Binomial Random WalkEach step moves up 1 unit or down 1 unit Each step moves up 1 unit or down 1 unit according to the random flip of a coinaccording to the random flip of a coin
00
-1-1
11
22
00
-1-1
11
22
00
-1-1
11
-1-1
-2-2
00
11
00
-1-1
11
22
+2+2
+2+2+2+2
-2-2
00
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HHTTHHTTHTHTHTHTHTTHHTTHTHHTTHHTTHTHTHTHTTHHTTHH
HHHTHHHTHHTHHHTHHTHHHTHHTHHHTHHH
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00
+2+2
+4+4
-2-2
-4-4
Number of ways Number of ways final value occursfinal value occurs
11
44
66
44
11
Distribution of Distribution of final valuesfinal values
The outcome depends The outcome depends upon the number of upon the number of Heads, not the order Heads, not the order in which they occurin which they occur
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-40
-30
-20
-10
0
10
20
30 Distribution of Distribution of final valuesfinal values
Excel Excel LinkLink
Word Word LinkLink
100 step random walks100 step random walks
22
66 66
1111 11
33 331111
111111
11 55
77 2121151511
353511
1155 11
2121 77111515
113535
66
1010 10104444
2020
11
Related to Row 101 Related to Row 101 of Pascal’s Triangleof Pascal’s Triangle
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Sunshine State Standards
MA.E.1.3.1 The student collects, organizes, and displays
data in a variety of forms, including tables, line graphs,
charts, bar graphs,to determine how different ways of presenting data can lead to different interpretations.
MA.E.1.3.3 The student uses technology, such as graphing calculators and computer spreadsheets, to
analyzedata and create graphs.
MA.E.2.3.1 The student compares experimental results with
mathematical expectations of probabilities.
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Food for Thought…
1. Why is random walk theory also called the gambler’s ruin theory?
2. Will a drunkard ever get back to his home from a city bar? Explain.
3. Would a “drunk” bird ever find its nest? Explain.
valcin_Rachelsummer07/quincunxmac
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Drunkard’s Path GIVEN 4 CHOICES