Andy and Becky are playing a guessing game. Andy takes a card from a standard 52-card deck. Becky is to guess the card, but she can ask one of the three questions below.
1) Is it a card red?
2) Is the card a face card (J, Q or K)?
3) Is the card an ace of spades?
Which question should Becky ask to maximise the probability of the guess being correct?
Becky may ask any of the above as her winning chance remains the same.
1) Is it a card red?
There is a (1/2) chance of getting a yes answer from Andy. If the answer is yes, Becky can guess 1 of 26 red cards, and if the answer is no, she can guess 1 of 26 black cards.
2) Is the card a face card (J, Q or K)?
The probability of Andy answering yes is 12/52, and no is 40/52. If yes is the answer, Becky can guess the correct one with a probability of (1/12), and for no, it is (1/40). So the overall probability is (12/52) x (1/12) + (40/52) x (1/40) = (1/52)+(1/52) = 1/26.
3) Is the card an ace of spades?
For the last question, the probability of yes is (1/52) and no is (51/52). Correct guessing probability if yes is 1 and if no is (1/51). The overall probability is (1/52) x 1 + (51/52) x (1/51) = 1/26.
