Yumeko's Gamble
Combinatorics · proposed by @sevenofhearts
Yumeko Jabami encounters a peculiar deck at Hyakkaou Private Academy. Each suit of this card contains infinitely many cards, numbered by every positive integer. Thus, the suit of hearts contains \(H_1, H_2, H_3, \ldots\), with no highest-numbered card.
The dealer challenged Yumeko. If she draws the seven of hearts, \(H_7\), she wins \(¥10^{12}\); otherwise, she wins nothing. Yumeko accepts the dealer's challenge. She draws exactly one card at random and is guaranteed that it belongs to the suit of hearts. The dealer claims that every heart has an equal probability of being drawn.
Assuming that the dealer was telling the truth, what is the probability that Yumeko wins?
A$0$
B$\frac{1}{52}$
C$1$
D$10^{-12}$
E$\nexists\,P$ satisfying the given conditions