There are two boxes. Box I contains one white card and two black cards and Box II contains one white card and a black card. Two persons P and Q play a game. P picks a card randomly from Box I. If P finds the white card, P wins and the game stops. If P finds the black card, Q draws a card randomly from Box II. If Q finds the white card, Q wins. The game stops whether Q draws the white card or the black card. Which one of the following is correct?
1. P draws a card from Box I (1 White, 2 Black).
* If P draws White (probability 1/3), P wins and the game stops.
* If P draws Black (probability 2/3), P loses this step, and Q draws from Box II.
2. If P drew Black, Q draws a card from Box II (1 White, 1 Black).
* If Q draws White (probability 1/2), Q wins. The game stops.
* If Q draws Black (probability 1/2), Q loses. The game stops.
Let’s analyze the outcomes:
* Outcome 1: P draws White (Prob=1/3). P wins.
* Outcome 2: P draws Black (Prob=2/3) AND Q draws White (Prob=1/2). Q wins. Probability = (2/3)*(1/2) = 1/3.
* Outcome 3: P draws Black (Prob=2/3) AND Q draws Black (Prob=1/2). Neither P nor Q wins. Probability = (2/3)*(1/2) = 1/3.
Now let’s evaluate the options:
A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct.
B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P’s winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect.
C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect.
D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1/3. So this statement is correct.