There are m white and n red marbles in a closed box.
Two players take turns on drawing randomly a marble, without returning it back.
The 1st player drawing a white marble wins.

Evaluate the probability that the first player wins.

Check your formula for (m,n)= (2,3) and (2,4).

m/(m+n) + (n/(m+n))*((n-1)/(m+n-1))*(m/(m+n-2)) + (n/(m+n))*((n-1)/(m+n-1))*((n-2)/(m+n-2))*((n-3)/(m+n-3))*(m/(m+n-4)) + ...

or in Sigma and Pi notation (Sum and Product):

Sum{i=0 to floor[n/2]} (m/(m+n-2*i)) * Prod{j=0 to 2*i-1}(n-j)/(m+n-j)

assuming that Prod will evaluate to 1 if 2*i-1 < 0, otherwise we'd have to take the first term of the original formula separately and start i off at 1 instead of zero. The Wikipedia article on Multiplication, in the section on Capital Pi notation states that for Pi{m to n}, "If m > n, the product is the empty product, with the value 1."

An alternative formulation is recursive:

p(a,0)=1

p(m,n) = m/(m+n) + (n/(m+n))*(1 - p(m,n-1)) when n>0,

as the first player can win on the first try, or, with the reciprocal probability, after the second player is given a shot but ultimately fails to win.

Both are implemented in:

DECLARE FUNCTION p2# (m#, n#)
DECLARE FUNCTION p1# (m#, n#)
CLEAR , , 25000
DEFDBL A-Z
CLS
FOR n = 0 TO 33
 PRINT n; p1(2, n), p2(2, n)
NEXT

FUNCTION p1 (m, n)
  tot = 0
  FOR i = 0 TO INT(n / 2)
    prod = m / (m + n - 2 * i)
    FOR j = 0 TO 2 * i - 1
      prod = prod * (n - j) / (m + n - j)
    NEXT j
    tot = tot + prod
  NEXT i
  p1 = tot
END FUNCTION

FUNCTION p2 (m, n)
  IF n = 0 THEN
   p = 1
  ELSE
   p = m / (m + n) + (n / (m + n)) * (1 - p2(m, n - 1))
  END IF
  p2 = p
END FUNCTION

All with m=2:

 n    Sigma Pi                 recursive
 0  1                        1
 1  .6666666666666666        .6666666666666666
 2  .6666666666666666        .6666666666666667
 3  .6                       .6
 4  .6                       .6
 5  .5714285714285714        .5714285714285714
 6  .5714285714285715        .5714285714285714
 7  .5555555555555555        .5555555555555556
 8  .5555555555555556        .5555555555555556
 9  .5454545454545454        .5454545454545454
 10  .5454545454545454       .5454545454545455
 11  .5384615384615384       .5384615384615384
 12  .5384615384615383       .5384615384615384
 13  .5333333333333333       .5333333333333333
 14  .5333333333333333       .5333333333333333
 15  .5294117647058824       .5294117647058824
 16  .5294117647058824       .5294117647058824
 17  .5263157894736842       .5263157894736842
 18  .5263157894736842       .5263157894736843
 19  .5238095238095238       .5238095238095237
 20  .5238095238095238       .5238095238095238
 21  .5217391304347825       .5217391304347826
 22  .5217391304347827       .5217391304347826
 23  .52                     .52
 24  .52                     .52
 25  .5185185185185185       .5185185185185185
 26  .5185185185185185       .5185185185185186
 27  .5172413793103449       .5172413793103448
 28  .5172413793103449       .5172413793103449
 29  .5161290322580645       .5161290322580645
 30  .5161290322580647       .5161290322580645
 31  .515151515151515        .5151515151515151
 32  .5151515151515152       .5151515151515151
 33  .5142857142857142       .5142857142857143
 

As given above, p(2,3) = p(2,4) = .6

Posted by Charlie on 2012-10-17 12:43:07