Answer
$\displaystyle \frac{1}{42}$
Work Step by Step
$P(E)=\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{n(E)}{n(S)}$
$n(S)=C(10,5)=252$
E = (4 out of 4 red marbles) AND (one chosen of the 6 remaining)
$n(E)=C(4,4)\cdot C(6,1)=6$
$P(E)=\displaystyle \frac{6}{252}=\frac{1}{42}$