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Sum up to 6

Posted by Chris on September 29, 2011 – 12:46 pm

Roll a dice die until you get a 6. On average, what will the sum of the rolls be before you get the 6?

e.g. if you rolled 1,2,5,6 the sum would be 8.

This post is under “MathsChallenge” and has 14 respond so far.
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14 Responds so far- Add one»

  1. 1. Wizard of Oz Said:

    Your chances of NOT getting a 6 after n throws is (5/6)^n.
    This becomes less than 50% after 4 throws, i.e. on average you will get 3 throws without a 6 before you get a 6.
    These 3 throws can each range from 1 to 5, averaging 3.
    So the average sum of rolls before getting a 6 is 3×3 = 9.

  2. 2. Chris Said:

    Hi Wiz. You are saying that on average you only need 4 rolls to get any given number.

    Take a look at to see what another Wizard of Oz said ;)

  3. 3. Jan Said:

    I’m going to say 15.

  4. 4. Chris Said:

    Hi Jan. Why 15?

  5. 5. Kel9902 Said:


  6. 6. ragknot Said:

    I got 21. Did I miss something?
    r-the roll#
    t=sub total
    a=count of sixes
    b=grand total

    Sub test()
    Dim a, b, t, r As Long
    For i = 1 To 10000000
    r = Int(Rnd() * 6 + 1)
    t = t + r
    If r = 6 Then
    a = a + 1
    b = b + t
    t = 0
    End If
    Next i
    Debug.Print b / a
    End Sub

    Result of b/a=20.9999999999

  7. 7. Jan Said:

    Basically i deduced the answer from the two posts above me.
    On average 6 rolls to get a 6, while not rolling a 6 the average roll is 3. 3×5=15.
    ragknot, i think your program adds the final 6 to the sum: 15+6=21.

  8. 8. Chris Said:

    Hi Jan. You’re right, 15 is the correct answer :)

    The average number of rolls to get a 6 is 6. So the average number for the previous 5 rolls is (1+2+3+4+5)/5 = 3, and of course the average total is 3*5 = 15.

    You’re also right, ragknot included the 6. Including the 6 makes the average roll be (1+2+3+4+5+6)/6 = 3.5, and 3.5*6 = 21 for the average total.

  9. 9. JB Said:

    Hi all. I don’t mean to be picayune, but the term for a single cube is ‘die’. I started looking at this problem in terms of a pair of dice and was coming up with completely different results. But my misunderstanding is a good basis for the question, what is the average sum when rolling a pair of dice until a six is rolled?

  10. 10. Chris Said:

    Hi JB. I goofed. I normally say “die”.

  11. 11. ragknot Said:

    yeah, someone was right, I should not have counted that last six, so 21 -6 would be 15.

  12. 12. Chris Said:

    Hi ragknot. I must have goofed again, I had written a reply along the lines of you had done the opposite if missing something. I must have forgotten to click submit.

  13. 13. JB Said:

    Just to be clear on the problem, I am talking about the sum of the 2 dice equaling 6.

  14. 14. Chris Said:

    Hi JB. I’ve put your problem (and more) up on a new page.

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