Question 375604
{{{C(200,40)*(1/6)^40*(5/6)^160}}}.  To get an approximate answer, use the normal approximation to the binomial distribution:
{{{P(X <= 40.5) = P((X - 100/3)/sqrt(250/9) <= (40.5 - 100/3)/sqrt(250/9)= (40.5 - 100/3)/5.27 = 1.36)}}} ; hence {{{P(X <= 40.5) = P(Z <= 1.36)}}}.
{{{P(X <= 39.5) = P((X - 100/3)/sqrt(250/9) <= (39.5 - 100/3)/sqrt(250/9)= (39.5 - 100/3)/5.27= 1.17)}}} ; hence {{{P(X <= 39.5) = P(Z <= 1.17)}}}.
Then the probability is approximately {{{P(Z <= 40.5) - P(Z <= 39.5)}}} = 0.9131 - 0.8790 = 0.0341.
(Here {{{mu = np = 100/3}}}, and {{{sigma = sqrt(npq) = sqrt(250/9)}}}.)