E[Y] = 11 ((11)/(120)) + (4)/(3)

$$E[Y] = 11 \left(\frac{11}{120}\right) + \frac{4}{3}$$ $$E[Y] = \frac{121}{120} + \frac{4}{3}$$

To add these fractions, find a common denominator, which is 120:

$$E[Y] = \frac{121}{120} + \frac{4 \times 40}{3 \times 40}$$ $$E[Y] = \frac{121}{120} + \frac{160}{120}$$ $$E[Y] = \frac{121 + 160}{120}$$ $$E[Y] = \frac{281}{120}$$

Now, calculate $E[2Y + 4]$ using the property $E[aY + b] = aE[Y] + b$:

$$E[2Y + 4] = 2E[Y] + 4$$

Substitute the value of $E[Y]$:

$$E[2Y + 4] = 2 \left(\frac{281}{120}\right) + 4$$ $$E[2Y + 4] = \frac{2 \times 281}{120} + 4$$ $$E[2Y + 4] = \frac{562}{120} + 4$$

Simplify the fraction $\frac{562}{120}$ by dividing both numerator and denominator by 2:

$$E[2Y + 4] = \frac{281}{60} + 4$$

Convert 4 to a fraction with a denominator of 60:

$$E[2Y + 4] = \frac{281}{60} + \frac{4 \times 60}{60}$$ $$E[2Y + 4] = \frac{281}{60} + \frac{240}{60}$$ $$E[2Y + 4] = \frac{281 + 240}{60}$$ $$E[2Y + 4] = \frac{521}{60}$$

The value of $E[2Y + 4]$ is $\boxed{\frac{521}{60}}$.