Question

Your friend has chosen a card from a standard deck of 52 playing cards and no one knows the card except himself. Now you have to guess the unknown card. Before guessing the card, you can ask your friend exactly one question, the question must be either Q1, Q2 or Q3 below:

Q1. whether the chosen card is an ace (A)?

Q2. whether the chosen card is a spade (♠)?

Q3. whether the chosen card is the ace of spades (A♠)?

Your friend will answer truthfully. What question would you prefer to ask so that it is more helpful to make a correct guess? Justify your answer.

Answer 1

It is possible to prove that no matter the question since there is only one question the probability that we would make a correct guess remains the same.

This is because for each question the friend will answer either yes or no and the probability of answering yes or no depends upon the total number of that group cards in the deck. Also for each case yes or no we need to calculate the probability of correct guessing.

P(friend saying yes) be denoted by P(y)

P(friend saying no) be denoted by P(n)

P(correct guess given yes) denoted by P(C/y)

P(correct guess given no) denoted by P(C/n)

P(correct guess) denoted by P(C)

Also P(C)=P(y)*P(C/y)+P(n)*P(C/n)

Now for Q1 whether ace since there are 4 aces in a deck.

P(y)=4/52 P(n)=48/52 since 4 aces in deck and he having ace is 4/52

P(C/y)=1/4 We know it is among 4 aces

P(C/n)=1/48 We know it is not any aces 1 among 48 others

So P(C)=4/52*1/4 + 48/52*1/48 = 1/52+1/52=2/52=1/26

Now suppose we ask Q2

Remember there are 13 spades in deck everything else same.

P(y)=13/52 P(n)=39/52 since 13 spades in deck and he having spades is 13/52

P(C/y)=1/13 We know it is among 4 aces

P(C/n)=1/39 We know it is not any aces 1 among 48 others

So P(C)=13/52*1/13 + 39/52*1/39 = 1/52+1/52=2/52=1/26

Now suppose we ask Q3 Only one ace of spades in deck

P(y)=1/52 P(n)=51/52 since 1 ace of spades

P(C/y)=1 We know it is ace of spades

P(C/n)=1/51

So P(C)=1/52*1 + 51/52*1/51 = 1/52+1/52=2/52=1/26

So since we only get one question whatever question we ask probability of us guessing corrrectly remains same 1/26.

It doesnt matter which question is asked.

Hope it is clear .

Happy learning.