Exercise 26 on page 39 of Introduction to Probability

Two cards are drawn successively from a deck of 52 cards. Find the probability that the second card is higher in rank than the first card. Hint: Show that 1 = \(P(higher) + P(lower) + P(same)\) and use the fact that \(P(higher) = P(lower)\)

Clearly, there are 3 possibilities when drawing two cards: either the second card is lower than the first, higher than the first, or equal to it (in rank). Therefore, \(P(higher) + P(lower) + P(same)\).

Assuming that \(P(higher) = P(lower)\) as stated in the problem, let us set the probabilities of those two values as \(x\). Therefore

\[1 = 2x + P(same)\]

\(P(same)\) is equal to the probability of the same card being drawn, so \(\frac{1}{52}\).

\[1 = 2x + \frac{1}{52}\] \[2x = 1 - \frac{1}{52}\] \[2x = \frac{51}{52}\] \[x = \frac{51}{104}\]

\[x = 49.0346%\]