1. **State the problem:** We want to find the probability that an 11-letter password, formed from the English alphabet letters at random, consists only of vowels. 2. **Identify vowels and total letters:** The English alphabet has 26 letters. The vowels are A, E, I, O, U, so there are 5 vowels. 3. **Total number of possible passwords:** Since each position can be any of the 26 letters, the total number of 11-letter passwords is $$26^{11}$$. 4. **Number of favorable passwords (all vowels):** Each position must be one of the 5 vowels, so the number of such passwords is $$5^{11}$$. 5. **Probability formula:** $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5^{11}}{26^{11}} = \left(\frac{5}{26}\right)^{11}$$ 6. **Calculate the probability:** $$\left(\frac{5}{26}\right)^{11} = \frac{5^{11}}{26^{11}}$$ 7. **Numerical evaluation:** Calculate $$\left(\frac{5}{26}\right)^{11} \approx 2.758924e-09$$ (rounded to 9 decimals). **Final answer:** $$\boxed{0.000000002759}$$