Problem J
Pull the Queen
Your friend invents a mysterious card game and wants you to bet all your money on it. Suspicious of his motives, you decide to analyze the game before agreeing. He also claims that if you win within a certain number of turns, you receive an even larger reward. The game uses a standard deck of 52 cards, containing 4 cards of each rank.
Objective
The goal is to pull a Queen from the deck. The player who pulls a Queen wins immediately.
Rules
One of you goes first. Call the first player Player A and the second player, Player B. The players alternate drawing cards from the deck until a Queen is drawn.
Each card has the following behavior:
|
Card |
Action |
|
King |
Return this card to the deck and reshuffle. |
|
Queen |
Game ends immediately. The player who drew it wins. |
|
Other |
Remove the card from the deck and continue. |
A turn consists of both players drawing once (i.e., one draw by Player A and one draw by Player B).
Input
The first line of input contains a single integer $N$ ($1 \leq N \leq 100$). Each of the next $N$ lines contains a case in the format “$p\ M$”, where:
-
$p \in \{ \texttt{A}, \texttt{B}\} $ indicates which player you are,
-
$1 \leq M \leq 50$ is the maximum number of turns allowed.
Output
For each test case, output the probability that you win the game by drawing a Queen in $M$ or fewer turns.
Your answer must have an absolute error of at most $10^{-10}$.
| Sample Input 1 | Sample Output 1 |
|---|---|
2 A 1 A 2 |
0.0769230769231 0.1447622487552 |
| Sample Input 2 | Sample Output 2 |
|---|---|
3 A 3 B 2 B 12 |
0.2042905382172 0.1358715967826 0.4414500410073 |
