perplexus.info :: Probability : Gambler's policy

Gambler's policy (Posted on 2016-05-20)

A gambler decides to play roulette. His initial investment is $200. He bets $100 each time and will quit when one of the following events happens:

i.He loses his money, within less than 8 bets.
or
ii. He's got $500, within less than 8 bets.
or
iii. He has played 8 times ending up with any amount.

a. What is the number of ways the betting can ocur?
b. What is the most likely length of his play (number of bets) within the above constraints?
c. Provide the probabilities of each of the three events (i, ii, iii).

Remark: Assume betting on even chances i.e. w/l=1

d. (bonus question) Would your answers change if the w/l odds were 18/19?

No Solution Yet Submitted by Ady TZIDON

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computer assisted solution

| Comment 1 of 2

With even probabilities these are the 88 ways in which results could happen:

0 2 -1 -1 0.25

0 4 -1 1 -1 -1 0.0625

0 6 -1 1 -1 1 -1 -1 0.015625

0* -1 1 -1 1 -1 1 -1 -1 0.00390625

2* -1 1 -1 1 -1 1 -1 1 0.00390625

2* -1 1 -1 1 -1 1 1 -1 0.00390625

4* -1 1 -1 1 -1 1 1 1 0.00390625

0* -1 1 -1 1 1 -1 -1 -1 0.00390625

2* -1 1 -1 1 1 -1 -1 1 0.00390625

2* -1 1 -1 1 1 -1 1 -1 0.00390625

4* -1 1 -1 1 1 -1 1 1 0.00390625

2* -1 1 -1 1 1 1 -1 -1 0.00390625

4* -1 1 -1 1 1 1 -1 1 0.00390625

5 -1 1 -1 1 1 1 1 0.0078125

0 6 -1 1 1 -1 -1 -1 0.015625

0* -1 1 1 -1 -1 1 -1 -1 0.00390625

2* -1 1 1 -1 -1 1 -1 1 0.00390625

2* -1 1 1 -1 -1 1 1 -1 0.00390625

4* -1 1 1 -1 -1 1 1 1 0.00390625

0* -1 1 1 -1 1 -1 -1 -1 0.00390625

2* -1 1 1 -1 1 -1 -1 1 0.00390625

2* -1 1 1 -1 1 -1 1 -1 0.00390625

4* -1 1 1 -1 1 -1 1 1 0.00390625

2* -1 1 1 -1 1 1 -1 -1 0.00390625

4* -1 1 1 -1 1 1 -1 1 0.00390625

5 -1 1 1 -1 1 1 1 0.0078125

0* -1 1 1 1 -1 -1 -1 -1 0.00390625

2* -1 1 1 1 -1 -1 -1 1 0.00390625

2* -1 1 1 1 -1 -1 1 -1 0.00390625

4* -1 1 1 1 -1 -1 1 1 0.00390625

2* -1 1 1 1 -1 1 -1 -1 0.00390625

4* -1 1 1 1 -1 1 -1 1 0.00390625

5 -1 1 1 1 -1 1 1 0.0078125

5 -1 1 1 1 1 0.03125

0 4 1 -1 -1 -1 0.0625

0 6 1 -1 -1 1 -1 -1 0.015625

0* 1 -1 -1 1 -1 1 -1 -1 0.00390625

2* 1 -1 -1 1 -1 1 -1 1 0.00390625

2* 1 -1 -1 1 -1 1 1 -1 0.00390625

4* 1 -1 -1 1 -1 1 1 1 0.00390625

0* 1 -1 -1 1 1 -1 -1 -1 0.00390625

2* 1 -1 -1 1 1 -1 -1 1 0.00390625

2* 1 -1 -1 1 1 -1 1 -1 0.00390625

4* 1 -1 -1 1 1 -1 1 1 0.00390625

2* 1 -1 -1 1 1 1 -1 -1 0.00390625

4* 1 -1 -1 1 1 1 -1 1 0.00390625

5 1 -1 -1 1 1 1 1 0.0078125

0 6 1 -1 1 -1 -1 -1 0.015625

0* 1 -1 1 -1 -1 1 -1 -1 0.00390625

2* 1 -1 1 -1 -1 1 -1 1 0.00390625

2* 1 -1 1 -1 -1 1 1 -1 0.00390625

4* 1 -1 1 -1 -1 1 1 1 0.00390625

0* 1 -1 1 -1 1 -1 -1 -1 0.00390625

2* 1 -1 1 -1 1 -1 -1 1 0.00390625

2* 1 -1 1 -1 1 -1 1 -1 0.00390625

4* 1 -1 1 -1 1 -1 1 1 0.00390625

2* 1 -1 1 -1 1 1 -1 -1 0.00390625

4* 1 -1 1 -1 1 1 -1 1 0.00390625

5 1 -1 1 -1 1 1 1 0.0078125

0* 1 -1 1 1 -1 -1 -1 -1 0.00390625

2* 1 -1 1 1 -1 -1 -1 1 0.00390625

2* 1 -1 1 1 -1 -1 1 -1 0.00390625

4* 1 -1 1 1 -1 -1 1 1 0.00390625

2* 1 -1 1 1 -1 1 -1 -1 0.00390625

4* 1 -1 1 1 -1 1 -1 1 0.00390625

5 1 -1 1 1 -1 1 1 0.0078125

5 1 -1 1 1 1 0.03125

0 6 1 1 -1 -1 -1 -1 0.015625

0* 1 1 -1 -1 -1 1 -1 -1 0.00390625

2* 1 1 -1 -1 -1 1 -1 1 0.00390625

2* 1 1 -1 -1 -1 1 1 -1 0.00390625

4* 1 1 -1 -1 -1 1 1 1 0.00390625

0* 1 1 -1 -1 1 -1 -1 -1 0.00390625

2* 1 1 -1 -1 1 -1 -1 1 0.00390625

2* 1 1 -1 -1 1 -1 1 -1 0.00390625

4* 1 1 -1 -1 1 -1 1 1 0.00390625

2* 1 1 -1 -1 1 1 -1 -1 0.00390625

4* 1 1 -1 -1 1 1 -1 1 0.00390625

5 1 1 -1 -1 1 1 1 0.0078125

0* 1 1 -1 1 -1 -1 -1 -1 0.00390625

2* 1 1 -1 1 -1 -1 -1 1 0.00390625

2* 1 1 -1 1 -1 -1 1 -1 0.00390625

4* 1 1 -1 1 -1 -1 1 1 0.00390625

2* 1 1 -1 1 -1 1 -1 -1 0.00390625

4* 1 1 -1 1 -1 1 -1 1 0.00390625

5 1 1 -1 1 -1 1 1 0.0078125

5 1 1 -1 1 1 0.03125

5 1 1 1 0.125

0.453125 .28125 .265625 1

0 .25 .125 .125 .09375 .078125 .0625 .265625

done

Lines beginning 0 or 5 represent cases where the gambler lost all his money, or he gained the extra $300 to wind up with $500; if this is not followed by an asterisk it was before bet #8. Other digits are followed by a * to indicate 8 bets took place; the digit represents how many hundred the gambler left with, which could be 0 or 5 even though that doesn't satisfy the requirement for event i or ii. The sequence of positive and negative 1's represent the outcome of each play, representing the win or loss in 100's. Finally the line shows the probability of this particular outcome sequence.

Events i, ii and iii have respective probabilities:

0.453125 .28125 .265625

as given above.

The respective probabilities of outcomes of: zero, $500 and stopping because neither has been achieved after 8 bets are:

0.50390625 .28125 .21484375 which do add to 1 as a check.

The probabilities of ending after 1 through 8 bets are respectively:

0 .25 .125 .125 .09375 .078125 .0625 .265625

The difference between the 0.453125 probability of event i, and the 0.50390625 of ending with no money is the result of event i not including going broke on the 8th bet.

With this even probability on each bet, the most likely number of rounds is 8, just barely beating out 2.

DefDbl A-Z

Dim crlf$, probLen(8), probNow, p, resources, hist(8), prob0, prob5, prob8

Private Sub Form_Load()

Form1.Visible = True

Text1.Text = ""

crlf = Chr$(13) + Chr$(10)

p = 0.5

probNow = 1: resources = 2 ' resources are in hundreds

spinIt 1

Text1.Text = Text1.Text & crlf & prob0 & Str(prob5) & Str(prob8) & " " & Str(prob0 + prob5 + prob8) & crlf

For i = 1 To 8

Text1.Text = Text1.Text & Str(probLen(i)) & " "

Text1.Text = Text1.Text & crlf

Text1.Text = Text1.Text & crlf & " done"

End Sub

Sub spinIt(wh)

For chg = -1 To 1 Step 2

saveProbNow = probNow

If chg = -1 Then probNow = probNow * (1 - p) Else probNow = probNow * p

hist(wh) = chg

resources = resources + chg

If wh = 8 Then

Text1.Text = Text1.Text & resources & "* "

For i = 1 To wh

Text1.Text = Text1.Text & Str(hist(i)) & " "

prob8 = prob8 + probNow

Text1.Text = Text1.Text & probNow & crlf

probLen(wh) = probLen(wh) + probNow

Else

Select Case resources

Case 0

Text1.Text = Text1.Text & resources & " " & wh & " "

For i = 1 To wh

Text1.Text = Text1.Text & Str(hist(i)) & " "

prob0 = prob0 + probNow

Text1.Text = Text1.Text & probNow & crlf

probLen(wh) = probLen(wh) + probNow

Case 5

Text1.Text = Text1.Text & resources & " "

For i = 1 To wh

Text1.Text = Text1.Text & Str(hist(i)) & " "

prob5 = prob5 + probNow

Text1.Text = Text1.Text & probNow & crlf

probLen(wh) = probLen(wh) + probNow

Case Else

spinIt wh + 1

End Select

End If

resources = resources - chg

probNow = saveProbNow

Next chg

End Sub

For the bonus, we just change p = 1 to p = 18/37, and we get:

0 2 -1 -1 0.263696128560993

0 4 -1 1 -1 -1 6.58758772592109E-02

0 6 -1 1 -1 1 -1 -1 1.64569393883493E-02

0* -1 1 -1 1 -1 1 -1 -1 4.11122956231954E-03

2* -1 1 -1 1 -1 1 -1 1 3.89484905903956E-03

2* -1 1 -1 1 -1 1 1 -1 3.89484905903956E-03

4* -1 1 -1 1 -1 1 1 1 3.68985700330064E-03

0* -1 1 -1 1 1 -1 -1 -1 4.11122956231954E-03

2* -1 1 -1 1 1 -1 -1 1 3.89484905903956E-03

2* -1 1 -1 1 1 -1 1 -1 3.89484905903956E-03

4* -1 1 -1 1 1 -1 1 1 3.68985700330064E-03

2* -1 1 -1 1 1 1 -1 -1 3.89484905903956E-03

4* -1 1 -1 1 1 1 -1 1 3.68985700330064E-03

5 -1 1 -1 1 1 1 1 7.18551100642756E-03

0 6 -1 1 1 -1 -1 -1 1.64569393883493E-02

0* -1 1 1 -1 -1 1 -1 -1 4.11122956231954E-03

2* -1 1 1 -1 -1 1 -1 1 3.89484905903956E-03

2* -1 1 1 -1 -1 1 1 -1 3.89484905903956E-03

4* -1 1 1 -1 -1 1 1 1 3.68985700330064E-03

0* -1 1 1 -1 1 -1 -1 -1 4.11122956231954E-03

2* -1 1 1 -1 1 -1 -1 1 3.89484905903956E-03

2* -1 1 1 -1 1 -1 1 -1 3.89484905903956E-03

4* -1 1 1 -1 1 -1 1 1 3.68985700330064E-03

2* -1 1 1 -1 1 1 -1 -1 3.89484905903956E-03

4* -1 1 1 -1 1 1 -1 1 3.68985700330064E-03

5 -1 1 1 -1 1 1 1 7.18551100642756E-03

0* -1 1 1 1 -1 -1 -1 -1 4.11122956231954E-03

2* -1 1 1 1 -1 -1 -1 1 3.89484905903956E-03

2* -1 1 1 1 -1 -1 1 -1 3.89484905903956E-03

4* -1 1 1 1 -1 -1 1 1 3.68985700330064E-03

2* -1 1 1 1 -1 1 -1 -1 3.89484905903956E-03

4* -1 1 1 1 -1 1 -1 1 3.68985700330064E-03

5 -1 1 1 1 -1 1 1 7.18551100642756E-03

5 -1 1 1 1 1 2.87630542918109E-02

0 4 1 -1 -1 -1 6.58758772592109E-02

0 6 1 -1 -1 1 -1 -1 1.64569393883493E-02

0* 1 -1 -1 1 -1 1 -1 -1 4.11122956231954E-03

2* 1 -1 -1 1 -1 1 -1 1 3.89484905903956E-03

2* 1 -1 -1 1 -1 1 1 -1 3.89484905903956E-03

4* 1 -1 -1 1 -1 1 1 1 3.68985700330064E-03

0* 1 -1 -1 1 1 -1 -1 -1 4.11122956231954E-03

2* 1 -1 -1 1 1 -1 -1 1 3.89484905903956E-03

2* 1 -1 -1 1 1 -1 1 -1 3.89484905903956E-03

4* 1 -1 -1 1 1 -1 1 1 3.68985700330064E-03

2* 1 -1 -1 1 1 1 -1 -1 3.89484905903956E-03

4* 1 -1 -1 1 1 1 -1 1 3.68985700330064E-03

5 1 -1 -1 1 1 1 1 7.18551100642756E-03

0 6 1 -1 1 -1 -1 -1 1.64569393883493E-02

0* 1 -1 1 -1 -1 1 -1 -1 4.11122956231954E-03

2* 1 -1 1 -1 -1 1 -1 1 3.89484905903956E-03

2* 1 -1 1 -1 -1 1 1 -1 3.89484905903956E-03

4* 1 -1 1 -1 -1 1 1 1 3.68985700330064E-03

0* 1 -1 1 -1 1 -1 -1 -1 4.11122956231954E-03

2* 1 -1 1 -1 1 -1 -1 1 3.89484905903956E-03

2* 1 -1 1 -1 1 -1 1 -1 3.89484905903956E-03

4* 1 -1 1 -1 1 -1 1 1 3.68985700330064E-03

2* 1 -1 1 -1 1 1 -1 -1 3.89484905903956E-03

4* 1 -1 1 -1 1 1 -1 1 3.68985700330064E-03

5 1 -1 1 -1 1 1 1 7.18551100642756E-03

0* 1 -1 1 1 -1 -1 -1 -1 4.11122956231954E-03

2* 1 -1 1 1 -1 -1 -1 1 3.89484905903956E-03

2* 1 -1 1 1 -1 -1 1 -1 3.89484905903956E-03

4* 1 -1 1 1 -1 -1 1 1 3.68985700330064E-03

2* 1 -1 1 1 -1 1 -1 -1 3.89484905903956E-03

4* 1 -1 1 1 -1 1 -1 1 3.68985700330064E-03

5 1 -1 1 1 -1 1 1 7.18551100642756E-03

5 1 -1 1 1 1 2.87630542918109E-02

0 6 1 1 -1 -1 -1 -1 1.64569393883493E-02

0* 1 1 -1 -1 -1 1 -1 -1 4.11122956231954E-03

2* 1 1 -1 -1 -1 1 -1 1 3.89484905903956E-03

2* 1 1 -1 -1 -1 1 1 -1 3.89484905903956E-03

4* 1 1 -1 -1 -1 1 1 1 3.68985700330064E-03

0* 1 1 -1 -1 1 -1 -1 -1 4.11122956231954E-03

2* 1 1 -1 -1 1 -1 -1 1 3.89484905903956E-03

2* 1 1 -1 -1 1 -1 1 -1 3.89484905903956E-03

4* 1 1 -1 -1 1 -1 1 1 3.68985700330064E-03

2* 1 1 -1 -1 1 1 -1 -1 3.89484905903956E-03

4* 1 1 -1 -1 1 1 -1 1 3.68985700330064E-03

5 1 1 -1 -1 1 1 1 7.18551100642756E-03

0* 1 1 -1 1 -1 -1 -1 -1 4.11122956231954E-03

2* 1 1 -1 1 -1 -1 -1 1 3.89484905903956E-03

2* 1 1 -1 1 -1 -1 1 -1 3.89484905903956E-03

4* 1 1 -1 1 -1 -1 1 1 3.68985700330064E-03

2* 1 1 -1 1 -1 1 -1 -1 3.89484905903956E-03

4* 1 1 -1 1 -1 1 -1 1 3.68985700330064E-03

5 1 1 -1 1 -1 1 1 7.18551100642756E-03

5 1 1 -1 1 1 2.87630542918109E-02

5 1 1 1 0.115136319665173

0.477732580021161 .258909570592026 .263357849386812 1

0 .263696128560993 .115136319665173 .131751754518422 8.62891628754327E-02 8.22846969417463E-02 5.74840880514205E-02 .263357849386812

done

where the E-02 represents *10^-2. So now, the probability of gambler's ruin in two bets slightly exceeds that of getting to bet number 8.

The number of ways of course remains the same, 88. Just the probabilities change.

Posted by Charlie on 2016-05-20 17:48:05

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