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 1 penny / 1 die (Posted on 2011-05-06)
A fair six sided die can roll any number from 1 to 6 with equal likelihood.
On fair coin, consider heads to have value 2 and tails to have value 1.

Consider the two experiments:

Experiment A: First roll the die. The outcome tells you how many times to flip the coin. x=the total value of the coin tosses.

Experiment B: First flip the coin. The outcome tells you how many times to roll the die. y=the total value of the die rolls.

1. Prove that the probability distributions of x and y are not the same.
2. How do the means of x and y compare?
3. How do the standard deviations of x and y compare?

 See The Solution Submitted by Jer No Rating

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 some solutions, but not sure about standard deviations | Comment 1 of 5

5    dim Prob(12)
10   for Die=1 to 6
20    PDie=1/6
30    for Tot=1 to 12
40      if Tot>=Die and Tot<=2*Die then
50         :PTot=PDie*combi(Die,Tot-Die)/2^Die
60         :Prob(Tot)=Prob(Tot)+PTot
70    next Tot
80   next Die
89   Sum=0
90   for I=1 to 12
100      print I,Prob(I);tab(22);Prob(I)/1;tab(45);:Cum=Cum+Prob(I)
101      for J=1 to int(150*Prob(I)):print "*";:next:print
102      Sum=Sum+I*Prob(I)
110   next
120   print Cum,Sum,Sum/1,
130   for I=1 to 12
140      Sumsq=(I-Sum)*(I-Sum)*Prob(I)
150   next I
160   Sd=sqrt(Sumsq)
170   print Sd
200   print
210   for I=1 to 12:Prob(I)=0:next I
310   for Coin=1 to 2
320    PCoin=1/2
330    for Die=1 to 6
335      if Coin=1 then
340       :Prob(Die)=Prob(Die)+PCoin/6
345      :else
350        :for Die2=1 to 6
355          :Prob(Die+Die2)=Prob(Die+Die2)+PCoin/36
360        :next Die2
365      :endif
370    next Die
380   next Coin
389   Sum=0:Cum=0:Sumsq=0
390   for I=1 to 12
400      print I,Prob(I);tab(22);Prob(I)/1;tab(45);:Cum=Cum+Prob(I)
401      for J=1 to int(150*Prob(I)):print "*";:next:print
402      Sum=Sum+I*Prob(I)
410   next
420   print Cum,Sum,Sum/1,
430   for I=1 to 12
440      Sumsq=(I-Sum)*(I-Sum)*Prob(I)
450   next I
460   Sd=sqrt(Sumsq)
470   print Sd

finds

Experiment A:

`total            probability                text graph 1       1/12         0.0833333333333333333 ************ 2       1/8          0.125                 ****************** 3       5/48         0.1041666666666666666 *************** 4       11/96        0.1145833333333333333 ***************** 5       7/64         0.109375              **************** 6       43/384       0.1119791666666666666 **************** 7       7/64         0.109375              **************** 8       13/128       0.1015625             *************** 9       5/64         0.078125              *********** 10      17/384       0.0442708333333333333 ****** 11      1/64         0.015625              ** 12      1/384        0.0026041666666666666             mean            s.d. 1       21/4   5.25    0.34445949507888442 `

Experiment B:

` 1       1/12         0.0833333333333333333 ************ 2       7/72         0.0972222222222222221 ************** 3       1/9          0.111111111111111111  **************** 4       1/8          0.125                 ****************** 5       5/36         0.1388888888888888888 ******************** 6       11/72        0.1527777777777777777 ********************** 7       1/12         0.0833333333333333333 ************ 8       5/72         0.0694444444444444444 ********** 9       1/18         0.0555555555555555555 ******** 10      1/24         0.0416666666666666666 ****** 11      1/36         0.0277777777777777777 **** 12      1/72         0.0138888888888888888 **             mean            s.d. 1       21/4   5.25    0.7954951288348659649 `

The means agree with what would be expected, as 1.5 * 3.5, which is the expected value of the coin tosses times the expected value on the die.

However, the standard deviation seems larger in experiment B while the data would seem to indicate it should be the other way around.

The calculations are in the program shown above.

 Posted by Charlie on 2011-05-06 14:12:53

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