Let x stand for the point at which the first player is indiffent
between standing on one turn or taking another. Let z be the
score on the first spin. The probability of winning by taking
another turn is the integral from z to 1 of t^{2} which
equals (1-z^{3})/3. By equating this with the probability
of winning by not taking another turn, z^{2}, you find
the indifference point: (1-z^{3})/3= z^{2}.
So the indifference point is x where x^{3}+x^{2}-1=0,
x =~ 0.53208889 .

The probability of winning is the integral from 0 to x of (1-z^{3})/3
plus the integral from x to 1 of z^{2}. After
integration this answer is (x - x^{4}/4)/3 + (1 -x ^{3})/3 =
(-x^{4}/4 -x^{3} + x + 1)/3 =~ .45380187 .

Michael Shackleford, A.S.A.