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 Flipping Coins (Posted on 2004-10-13)
You play a coin flipping game with 5 coins. On round 1 you flip all of them. On round 2, you pick up all the ones that came up tails (leaving all the heads alone) and flip them again. You continue to do this until all the coins are heads. For example:
```Round 1:  H T T H T
Round 2:  - H T - H
Round 3:  - - T - -
Round 4:  - - T - -
Round 5:  - - H - -
```
Done in 5 Rounds.

What is the expected number of rounds you'll need to finish the game?
What is the probability you will finish the game in 3 rounds or less?

 See The Solution Submitted by Brian Smith Rating: 3.7143 (7 votes)

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<TABLE style="WIDTH: 825pt; BORDER-COLLAPSE: collapse" cellSpacing=0 cellPadding=0 width=1099 border=0 x:str> <COLGROUP> <COL style="WIDTH: 56pt; mso-width-source: userset; mso-width-alt: 1894" width=74> <COL style="WIDTH: 73pt; mso-width-source: userset; mso-width-alt: 2483" width=97> <COL style="WIDTH: 68pt; mso-width-source: userset; mso-width-alt: 2329" width=91> <COL style="WIDTH: 24pt; mso-width-source: userset; mso-width-alt: 819" width=32> <COL style="WIDTH: 66pt" span=2 width=88> <COL style="WIDTH: 24pt; mso-width-source: userset; mso-width-alt: 819" width=32> <COL style="WIDTH: 66pt" span=2 width=88> <COL style="WIDTH: 24pt; mso-width-source: userset; mso-width-alt: 819" width=32> <COL style="WIDTH: 66pt" span=2 width=88> <COL style="WIDTH: 24pt; mso-width-source: userset; mso-width-alt: 819" width=32> <COL style="WIDTH: 70pt; mso-width-source: userset; mso-width-alt: 2380" width=93> <COL style="WIDTH: 66pt" width=88> <TBODY> <TR style="HEIGHT: 21pt" height=28> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 56pt; BORDER-BOTTOM: #c0c0c0; HEIGHT: 21pt; BACKGROUND-COLOR: #99ccff" width=74 height=28>

</TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 73pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=97>1 Coin</TD> <TD class=xl26 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 68pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=91> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 24pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=32> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88>2 Coins</TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 24pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=32> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88>3 Coins</TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 24pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=32> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88>4 Coins</TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88> </TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 24pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=32> </TD> <TD class=xl31 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 70pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=93>5 Coins</TD> <TD class=xl25 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; WIDTH: 66pt; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff" width=88>trial</TD></TR> <TR style="HEIGHT: 21.75pt" height=29> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; HEIGHT: 21.75pt; BACKGROUND-COLOR: #99ccff" height=29>Flips</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">n</TD> <TD class=xl30 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">P(n)</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">n</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">P(n)</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">n</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">P(n)</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">n</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">P(n)</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl32 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">n</TD> <TD class=xl29 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: windowtext 1pt solid; BACKGROUND-COLOR: #99ccff">P(n)</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num>1</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=1/(POWER(2,1))">0.5</TD> <TD class=xl28 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=B3">0.500000</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=1/POWER(2,2)">0.25</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=E3">0.25</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.125" x:fmla="=1/POWER(2,3)">0.125</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.125" x:fmla="=H3">0.125</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="6.25E-2" x:fmla="=1/POWER(2,4)">0.0625</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="6.25E-2" x:fmla="=K3">0.0625</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl33 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.125E-2" x:fmla="=1/POWER(2,5)">0.0312500</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.125E-2" x:fmla="=N3">0.03125</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A3+1">2</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=1/(POWER(2,A4))">0.25</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num x:fmla="=B4+C3">0.750000</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.3125" x:fmla="=(1/POWER(2,A4))+(1/4)*E3">0.3125</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.5625" x:fmla="=F3+E4">0.5625</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.296875" x:fmla="=(3/POWER(2,A4+2))+(3*E3/8)+ (1/8)*H3">0.296875</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.421875" x:fmla="=I3+H4">0.421875</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.25390625" x:fmla="=(4/(POWER(2,A4+3))+(6*E3/16)+(4*H3/16))+(1/16)*K3">0.253906</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.31640625" x:fmla="=L3+K4">0.316406</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.2060546875" x:fmla="=(5/(POWER(2,A4+4))+(10*E3/32)+(10*H3/32))+(5*K3/32)+(1/32)*N3">0.2060547</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.2373046875" x:fmla="=O3+N4">0.237305</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A4+1">3</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.125" x:fmla="=1/(POWER(2,A5))">0.125</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.875" x:fmla="=B5+C4">0.875000</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.203125" x:fmla="=(1/POWER(2,A5))+(1/4)*E4">0.203125</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.765625" x:fmla="=F4+E5">0.765625</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.248046875" x:fmla="=(3/POWER(2,A5+2))+(3*E4/8)+ (1/8)*H4">0.248047</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.669921875" x:fmla="=I4+H5">0.669922</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.269775390625" x:fmla="=(4/(POWER(2,A5+3))+(6*E4/16)+(4*H4/16))+(1/16)*K4">0.269775</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.586181640625" x:fmla="=L4+K5">0.586182</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.275604248046875" x:fmla="=(5/(POWER(2,A5+4))+(10*E4/32)+(10*H4/32))+(5*K4/32)+(1/32)*N4">0.2756042</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.512908935546875" x:fmla="=O4+N5">0.512909</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A5+1">4</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="6.25E-2" x:fmla="=1/(POWER(2,A6))">0.0625</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.9375" x:fmla="=B6+C5">0.937500</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.11328125" x:fmla="=(1/POWER(2,A6))+(1/4)*E5">0.113281</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.87890625" x:fmla="=F5+E6">0.878906</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.154052734375" x:fmla="=(3/POWER(2,A6+2))+(3*E5/8)+ (1/8)*H5">0.154053</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.823974609375" x:fmla="=I5+H6">0.823975</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.1862945556640625" x:fmla="=(4/(POWER(2,A6+3))+(6*E5/16)+(4*H5/16))+(1/16)*K5">0.186295</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.7724761962890625" x:fmla="=L5+K6">0.772476</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.21128749847412109" x:fmla="=(5/(POWER(2,A6+4))+(10*E5/32)+(10*H5/32))+(5*K5/32)+(1/32)*N5">0.2112875</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.72419643402099609" x:fmla="=O5+N6">0.724196</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A6+1">5</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.125E-2" x:fmla="=1/(POWER(2,A7))">0.03125</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.96875" x:fmla="=B7+C6">0.968750</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="5.95703125E-2" x:fmla="=(1/POWER(2,A7))+(1/4)*E6">0.05957</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.9384765625" x:fmla="=F6+E7">0.938477</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="8.5174560546875E-2" x:fmla="=(3/POWER(2,A7+2))+(3*E6/8)+ (1/8)*H6">0.085175</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.909149169921875" x:fmla="=I6+H7">0.909149</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.10826206207275391" x:fmla="=(4/(POWER(2,A7+3))+(6*E6/16)+(4*H6/16))+(1/16)*K6">0.108262</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.88073825836181641" x:fmla="=L6+K7">0.880738</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.12901875376701355" x:fmla="=(5/(POWER(2,A7+4))+(10*E6/32)+(10*H6/32))+(5*K6/32)+(1/32)*N6">0.1290188</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.85321518778800964" x:fmla="=O6+N7">0.853215</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A7+1">6</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="1.5625E-2" x:fmla="=1/(POWER(2,A8))">0.015625</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.984375" x:fmla="=B8+C7">0.984375</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.0517578125E-2" x:fmla="=(1/POWER(2,A8))+(1/4)*E7">0.030518</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.968994140625" x:fmla="=F7+E8">0.968994</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="4.4704437255859375E-2" x:fmla="=(3/POWER(2,A8+2))+(3*E7/8)+ (1/8)*H7">0.044704</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.95385360717773438" x:fmla="=I7+H8">0.953854</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="5.8211386203765869E-2" x:fmla="=(4/(POWER(2,A8+3))+(6*E7/16)+(4*H7/16))+(1/16)*K7">0.058211</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.93894964456558228" x:fmla="=L7+K8">0.93895</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="7.1063368581235409E-2" x:fmla="=(5/(POWER(2,A8+4))+(10*E7/32)+(10*H7/32))+(5*K7/32)+(1/32)*N7">0.0710634</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.92427855636924505" x:fmla="=O7+N8">0.924279</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A8+1">7</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="7.8125E-3" x:fmla="=1/(POWER(2,A9))">0.0078125</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.9921875" x:fmla="=B9+C8">0.992188</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="1.544189453125E-2" x:fmla="=(1/POWER(2,A9))+(1/4)*E8">0.015442</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.98443603515625" x:fmla="=F8+E9">0.984436</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="2.2891521453857422E-2" x:fmla="=(3/POWER(2,A9+2))+(3*E8/8)+ (1/8)*H8">0.022892</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.9767451286315918" x:fmla="=I8+H9">0.976745</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.0164662748575211E-2" x:fmla="=(4/(POWER(2,A9+3))+(6*E8/16)+(4*H8/16))+(1/16)*K8">0.030165</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.96911430731415749" x:fmla="=L8+K9">0.969114</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.7264545419020578E-2" x:fmla="=(5/(POWER(2,A9+4))+(10*E8/32)+(10*H8/32))+(5*K8/32)+(1/32)*N8">0.0372645</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.96154310178826563" x:fmla="=O8+N9">0.961543</TD></TR> <TR style="HEIGHT: 19.5pt" height=26> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; HEIGHT: 19.5pt; BACKGROUND-COLOR: #33cccc" align=right height=26 x:num x:fmla="=A9+1">8</TD> <TD class=xl27 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: #c0c0c0; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="3.90625E-3" x:fmla="=1/(POWER(2,A10))">0.0039063</TD> <TD class=xl24 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.99609375" x:fmla="=B10+C9">0.996094</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="7.7667236328125E-3" x:fmla="=(1/POWER(2,A10))+(1/4)*E9">0.007767</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.9922027587890625" x:fmla="=F9+E10">0.992203</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="1.1581838130950928E-2" x:fmla="=(3/POWER(2,A10+2))+(3*E9/8)+ (1/8)*H9">0.011582</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.98832696676254272" x:fmla="=I9+H10">0.988327</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="1.5352007234469056E-2" x:fmla="=(4/(POWER(2,A10+3))+(6*E9/16)+(4*H9/16))+(1/16)*K9">0.015352</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.98446631454862654" x:fmla="=L9+K10">0.984466</TD> <TD class=xl22 style="BORDER-RIGHT: #c0c0c0; BORDER-TOP: #c0c0c0; BORDER-LEFT: #c0c0c0; BORDER-BOTTOM: #c0c0c0; BACKGROUND-COLOR: #99ccff"> </TD> <TD class=xl34 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext 0.5pt solid; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="1.9077641219155339E-2" x:fmla="=(5/(POWER(2,A10+4))+(10*E9/32)+(10*H9/32))+(5*K9/32)+(1/32)*N9">0.0190776</TD> <TD class=xl23 style="BORDER-RIGHT: windowtext 0.5pt solid; BORDER-TOP: windowtext; BORDER-LEFT: windowtext; BORDER-BOTTOM: windowtext 0.5pt solid; BACKGROUND-COLOR: #33cccc" align=right x:num="0.98062074300742097" x:fmla="=O9+N10">0.980621</TD></TR></TBODY></TABLE>

These are the expected values of the problem that I found for each coin scenario up to 5 coins. Microsoft Excel is an excellent tool for recursive formulas any most other calculations of these types.

This was a cool problem that I enjoyed working on!

Michael

 Posted by Michael on 2004-11-21 16:43:29

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