What question to ask to maximize my chances of guessing the right card? I have a shuffled deck of cards and draw one randomly and put it face down. Another person looks at the card. I can ask them only one of the two questions: a) is the card red? b) is the card 10 of spades?
Which of the two questions should I ask to maximize my chances of guessing the correct card? I'm not sure if it's a trick question, because it seems obvious I should ask the first question. But I might be missing something major here. A friend of mine claims that both questions are equally good since they result in the same probability for guessing right: $1/26$
 A: The best way to see this is to ask yourself what can happen in each scenario. Suppose you asked the first question, about if the card was red or not.
If the card holder says yes, you now only have $26$ cards to guess from (a $\frac{1}{26}$ chance of guessing correctly). If the holder says no, you also have $26$ cards to guess from (because you would know it's a black card, also with $\frac{1}{26}$ chance of success).
Combining these probabilities, we have that $$P({\text{correct guess}})=\frac{1}{2}\left(\frac{1}{26}\right)+\frac{1}{2}\left(\frac{1}{26}\right)=\frac{1}{26}.$$In other words, either the card holder reveals that it's a red card AND (multiplication) you guess correctly, OR (addition) it's a black card AND you guess correctly.
Apply this same thinking to the second question: what happens if the holder says yes, it's a $10$ of spades? What are the chances of them saying that? What are the chances you guess the correct card if they say that? What if they said no? You should see that you also arrive at $P({\text{correct guess}})=\frac{1}{26}.$
