Win/Loss Probability and Random Process Simulation
2015-02-04 02:19
This GeoGebra applet demonstrates a mathematical model for win/loss probabilities, involving state transitions and random walks in a game. Using interactive controls to adjust the number of turns, win rate, and loss rate, it dynamically calculates and visualizes the number of surviving and eliminated paths along with their probability formulas. The model intuitively presents related concepts in pr
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<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelOffset x="-27" y="-11"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val="SetValue[Picker, ChoicePFormula]"/>
<coords x="-1.98" y="-1.98" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="2"/>
<caption val="機率算法"/>
</element>
<command name="Sequence">
<input a0="(((Binomial[nLose + (2 * k), k] - (2 * Binomial[nLose + (2 * k) - 1, k - 1])) * p^(k)) * q^(nLose + k))" a1="k" a2="0" a3="x1"/>
<output a0="nDeadP"/>
</command>
<element type="list" label="nDeadP">
<objColor r="255" g="0" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sum">
<input a0="nDeadP"/>
<output a0="pDead"/>
</command>
<element type="numeric" label="pDead">
<value val="0.34375"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sum">
<input a0="nAliveP"/>
<output a0="pAlive"/>
</command>
<element type="numeric" label="pAlive">
<value val="0.65625"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="txtPAlive" exp=""存活機率：" + pAlive + """ />
<element type="text" label="txtPAlive">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<absoluteScreenLocation x="58" y="531"/>
</element>
<expression label="txtPDead" exp=""敗亡機率：" + pDead + """ />
<element type="text" label="txtPDead">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<absoluteScreenLocation x="59" y="559"/>
</element>
<element type="point" label="OffsetDeadF">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.036963314750198" y="0.6805291357296513" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text["C^{" + (nLose + (2 * k)) + "}_{" + (k) + "}-2C^{" + (nLose + (2 * k) - 1) + "}_{" + (k - 1) + "}", Element[endsDead, k + 1] + OffsetDeadF, true, true]" a1="k" a2="0" a3="x1"/>
<output a0="txtWaysDeadFormula"/>
</command>
<element type="list" label="txtWaysDeadFormula">
<show object="true" label="false"/>
<condition showObject="Picker ≟ ChoiceFormula"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
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<element type="point" label="OffsetDeadP">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
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<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.8579556701322948" y="0.22306515503945626" z="1.0"/>
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<command name="Sequence">
<input a0="Text[Element[nDeadP, k], Element[endsDead, k] + OffsetDeadP]" a1="k" a2="1" a3="x1 + 1"/>
<output a0="txtDeadP"/>
</command>
<element type="list" label="txtDeadP">
<show object="true" label="false"/>
<condition showObject="Picker ≟ ChoiceP"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<font serif="false" sizeM="1.4" size="5" style="2"/>
</element>
<command name="Sequence">
<input a0="Text["(C^{" + nTurns + "}_{" + (k) + "}-C^{" + nTurns + "}_{" + (nTurns - nLose - k) + "})p^{" + (k) + "}q^{" + (nTurns - k) + "}", Element[endsTie, k - x1] + OffsetAliveF, true, true]" a1="k" a2="x1 + 1" a3="nTurns"/>
<output a0="txtAlivePFormula"/>
</command>
<element type="list" label="txtAlivePFormula">
<show object="true" label="false"/>
<condition showObject="Picker ≟ ChoicePFormula"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
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<pointStyle val="-1"/>
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<element type="point" label="OffsetDeadPF">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.3706783973302494" y="0.6606393974387756" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="txtQ" exp=""( q=1-p )""/>
<element type="text" label="txtQ">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="77" y="472"/>
</element>
<command name="Sequence">
<input a0="Text["(C^{" + (nLose + (2 * k)) + "}_{" + (k) + "}-2C^{" + (nLose + (2 * k) - 1) + "}_{" + (k - 1) + "})p^{" + (k) + "}q^{" + (nLose + k) + "}", Element[endsDead, k + 1] + OffsetDeadPF, true, true]" a1="k" a2="0" a3="x1"/>
<output a0="txtDeadPFormula"/>
</command>
<element type="list" label="txtDeadPFormula">
<show object="true" label="false"/>
<condition showObject="Picker ≟ ChoicePFormula"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="point" label="ChoiceNone">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="2"/>
<labelOffset x="-9" y="-11"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val="SetValue[Picker, ChoiceNone]"/>
<coords x="-1.0" y="-1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="2"/>
<caption val="無"/>
</element>
<command name="Sequence">
<input a0="Sequence[Vector[(j, k), (j + 1, k)], j, Max[0, k - nLose + 1], Min[nTurns - k - 1, k + nWin - 1]]" a1="k" a2="0" a3="y(A)"/>
<output a0="goRight"/>
</command>
<element type="list" label="goRight">
<show object="true" label="false"/>
<objColor r="153" g="204" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Sequence[Vector[(k, j), (k, j + 1)], j, Max[0, k - nWin + 1], Min[k + nLose - 1, nTurns - k - 1]]" a1="k" a2="0" a3="nTurns - 1"/>
<output a0="goUp"/>
</command>
<element type="list" label="goUp">
<show object="true" label="false"/>
<objColor r="255" g="192" b="203" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="(k, nTurns - k)" a1="k" a2="x1 + 1" a3="x2"/>
<output a0="endsTieFrame"/>
</command>
<element type="list" label="endsTieFrame">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="6"/>
<pointStyle val="5"/>
</element>
<expression label="x3" exp="floor(x(C))" />
<element type="numeric" label="x3">
<value val="6.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="(k, k - nWin)" a1="k" a2="nWin" a3="x3"/>
<output a0="endsWin"/>
</command>
<element type="list" label="endsWin">
<show object="true" label="false"/>
<objColor r="0" g="204" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="text6" exp=""淨勝數""/>
<element type="text" label="text6">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="18" y="134"/>
</element>
<element type="button" label="button5">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<bgColor r="153" g="204" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="25" y="153"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[nWin, nWin-1]"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<file name="/geogebra/gui/images/go-previous.png"/>
</element>
<element type="button" label="button6">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<bgColor r="153" g="204" b="255" alpha="255"/>
<layer val="2"/>
<labelOffset x="180" y="153"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[nWin, nWin+1]"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<file name="/geogebra/gui/images/go-next.png"/>
</element>
<expression label="txtEquWin" exp=""x-y=" + (LaTeX[nWin]) + """ />
<element type="text" label="txtEquWin">
<show object="false" label="false"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="C"/>
</element>
<expression label="txtEquLose" exp=""y-x=" + (LaTeX[nLose]) + """ />
<element type="text" label="txtEquLose">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="2.5000000000000036" y="9.400000000000004" z="1.0"/>
</element>
<expression label="txtEquTurn" exp=""x+y=" + (LaTeX[nTurns]) + """ />
<element type="text" label="txtEquTurn">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="1.5000000000000033" y="8.280000000000003" z="1.0"/>
</element>
<command name="RotateText">
<input a0="txtEquWin" a1="45*°"/>
<output a0="txtEquWinRot"/>
</command>
<element type="text" label="txtEquWinRot">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="C"/>
<labelOffset x="24" y="-76"/>
</element>
<command name="RotateText">
<input a0="txtEquLose" a1="45*°"/>
<output a0="txtEquLoseRot"/>
</command>
<element type="text" label="txtEquLoseRot">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="A + (0, 1)"/>
<labelOffset x="-8" y="-45"/>
</element>
<command name="RotateText">
<input a0="txtEquTurn" a1="-45*°"/>
<output a0="txtEquTurnRot"/>
</command>
<element type="text" label="txtEquTurnRot">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="(0, nTurns)"/>
<labelOffset x="-1" y="-23"/>
</element>
</construction>
</geogebra>