2 Drawn Cards: Higher Card
Two friends each draw one card (in succession, without replacement) from a standard 52‐card deck. The first friend wins if their card is strictly higher in rank than the second friend’s card, and otherwise the second friend wins. What is the probability that the first friend wins?
Answer
The probability that the two cards match in rank (i.e. a tie) is \(\tfrac{1}{17}\).
Therefore, with probability \(\tfrac{16}{17}\) the cards differ in rank.
By symmetry, in those cases where they differ, each friend’s card is equally likely to be the higher one. Hence the first friend wins in half of those scenarios.
Combining these points, the probability that the first friend’s card is strictly higher is
\[
\frac{1}{2} \;\times\; \frac{16}{17}
\;=\;
\frac{8}{17}
\;\approx\; 0.4706
\;\text{(about }47.06\%\text{).}
\]