HELP!!!! How many games of War would you expect to play until you had a game that lasted just 26 turns with no mini “Wars,” like the girl claimed?

Question

A girl claimed that she once won a game of War that lasted exactly 26 turns.

War is a two-player card game in which a standard deck of cards is first shuffled and then divided into two piles with 26 cards each — one pile for each player. In every turn of the game, both players flip over and reveal the top card of their deck.

The player whose card has a higher rank wins the turn and places both cards on the bottom of their pile. In the event that both cards have the same rank, the rules get a little more complicated, with each player flipping over additional cards to compare in a mini “War” showdown.

The girl said that for every turn of the game, she always drew the card of higher rank, with no mini “Wars.”

Assuming a deck is randomly shuffled before every game, how many games of War would you expect to play until you had a game that lasted just 26 turns with no mini “Wars,” like the girl claimed?

Answer 1

If ties weren't possible, the probability of winning a flip would be 1/2, and the probability of winning 26 flips would be 1/2^26, so we would expect to play 1/(1/2^26)) = 2^26 games.

However, as we have 4 suits, P(winning a flip) is reduced to 8/17

(It would be 1/2 - 1/2 P(tie), and P(tie) = 3 matches out of 51 remaining cards = 3/51 = 1/17, and 1/2 - 1/2(1/17) = 8/17).

Clearly, matches are not independent - if you have one match, you are more likely to have a second. However, we can still use this as an approximation.

1/(8/17)^26 = (17/8)^ 26