A basket has $6$ apples, $3$ mangoes, and $5$ oranges. One fruit is selected at random. What is the probability that it is neither an apple nor a mango?

First, let's find the total number of fruits in the basket:
$$\text{Total Fruits}=6\text{ apples}+3\text{ mangoes}+5\text{ oranges}=14\text{ fruits}$$

The condition requires the fruit to be "neither an apple nor a mango". This leaves only oranges.

$$\text{Favorable Outcomes (Oranges)}=5$$

$$\text{Probability}=\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}=\frac{5}{14}$$

Therefore, the probability is $\frac{5}{14}$.