  The celebrated Erd\H{o}s-Ko-Rado theorem shows that for $n \ge 2k$ the
largest intersecting $k$-uniform set family on $[n]$ has size
$\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set
families must be. Katona, Katona and Katona introduced the notion of most
probably intersecting families, which maximise the probability of random
subfamilies being intersecting.
  We study the most probably intersecting problem for $k$-uniform set families.
We provide a rough structural characterisation of the most probably
intersecting families and, for families of particular sizes, show that the
initial segment of the lexicographic order is optimal.
