The probability that you will choose a lemon-lime for your friend is the number of lemon-limes divided by the total number of drinks, 4/11.
The probability that you will subsequently choose one for yourself is the same ratio with different numbers (because a lemon-lime has already been selected), 3/10.
The probability of both of these events occurring is the product of their individual probabilities: (4/11)·(3/10) = 6/55