Question: Consider a deck of $52$ cards, ordered such that $A>K>Q>⋯>2$. I pick one first, then you pick one,
Suppose one suit is a trump suit. What is the probability that the second card would beat the first card in a trick? (i.e. it is a trump and the first card is not, or it is higher than the first card and in the same suit.)
My attempt:
Suppose we want to pick a trump suit at our second card and first card is not a trump. It has probability $\frac{13}{52}\times \frac{39}{51}.$
Now, if we do not want to pick a trump suit but it is higher than the first card and in the same suit, then the probability is $\frac{39}{52}\times \frac{?}{51}.$
I do not know what to input for $?$.
Best Answer
Given that the two cards are the same suit, the chance the second wins is $\frac 12$ by symmetry. Compute the chance that the two cards are the same suit (including trumps) and divide by $2$.