What is the probability that the committee includes at least 2 ladies?

Calculation:

Given,

A committee of 6 members is formed from a group of 7 gentlemen and 4 ladies.

The total number of ways to form the committee is:

\( \binom{11}{6} = 462 \)

The favorable outcomes (at least 2 ladies) are calculated by the sum of the cases:

\( \binom{4}{2} \times \binom{7}{4} + \binom{4}{3} \times \binom{7}{3} + \binom{4}{4} \times \binom{7}{2} \)

This simplifies to:

\( \binom{4}{2} \times \binom{7}{4} = 6 \times 35 = 210 \)

\( \binom{4}{3} \times \binom{7}{3} = 4 \times 35 = 140 \)

\( \binom{4}{4} \times \binom{7}{2} = 1 \times 21 = 21 \)

The total favorable outcomes are:

\( 210 + 140 + 21 = 371 \)

The probability is:

\( P(E) = \frac{371}{462} = \frac{53}{66} \)

Hence, the correct answer is Option 4.