A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or is a nail

$\frac{11}{16}$

Let A be the event that the item chosen is rusted and B be the event that the item chosen is a nail.

$∴ P(A)=\frac{8}{16}, P(B) = \frac{6}{16}$ and $ P(A ∩ B) = \frac{3}{16}$

Required probability $= P(A ∪ B) = P(A) + P(B) - P(A ∩ B)=\frac{8}{16}+\frac{6}{16}-\frac{3}{16}=\frac{11}{16}.$