# Is it better to ask a question or guess with 3 characters left in Guess Who?

Say you have 3 potential characters left.

If you go with a guess, you have a 1/3 chance that turn and a 1/2 chance next turn.

If you go with a hint, you would have a 50% chance of leaving 2 left (with a 50% chance to win next turn) or a 50% chance of leaving 1 left (with a 100% chance to win next turn).

I don't know statistics well enough to figure it out from there.

EDIT: We were not aware of the rule that state if you guess incorrectly it means you instantly lose. That definitely makes it clear which you should do. We will definitely try to follow that going forward, thanks all!

• When I played this as a child we would deliberately flip down down non eliminated characters to trick opponents into guessing or we would leave eliminated characters stood up so opponent didn’t know how close were to winning. As an adult this might have been against printed rules but was very much a house rule for us. – StartPlayer Jun 16 '20 at 7:06
• If you guess and get it wrong you instantly lose. Your question suggests you get another go which is not correct. Source of rules here. winning-moves.com/images/guesswho%20rules.pdf “ If you guess correctly – or your opponent guesses incorrectly – you win the game!” – StartPlayer Jun 16 '20 at 7:14
• We'd need clarification from OP, but I assume he is playing without the rule of "if you guess wrong you instantly lose." – Parseltongue Jun 16 '20 at 17:43

My original answer assumed that there was no penalty for guessing incorrectly. This answer assumes that you are playing with the rule that incorrect guesses cause a game loss. My original answer is provided below this one

## Never guess, unless a player has only a single character left

If your opponent guesses, then your odds of winning are 50% if they have two characters left or higher if they have more. If you guess, your odds are 50% if you have two characters left, or lower if you have more. Therefore, there is never any benefit to guessing - the odds are always equal to or better if your opponent guesses. This only changes if your opponent have a single character left. At that point the odds of losing jump to 100%, and any odds are better than that, so guess. (Also, if you only have 1 character left you should obviously guess as well)

## Guess. Always guess when there are three or fewer characters left

Any question you ask is guaranteed to divide the remaining characters into a group of one and a group of two.

Guessing also divides the characters into a group of one and a group of two.

Literally the only difference is that if the character is in the group of one and you ask then you need to take an additional turn to guess, whereas if you guess and the character is in the group of one then you've already won.

Your error is in assuming that asking a question has a 50% chance of putting the character in the group of one. It's actually 1/3, same as when you guess.

• Guessing doesn’t divide the group into 2 if wrong as guessing wrong is instant loss. – StartPlayer Jun 16 '20 at 7:15
• @StartPlayer fixed – Arcanist Lupus Jun 16 '20 at 12:36
• Assuming that asking a question has a 50% chance of putting the character into a group of 1, wasn't an "error" of my answer. That was an explicit assumption that was built into the OP's question. – Parseltongue Jun 16 '20 at 17:28
• @Parseltongue just because the OP stated the odds were 50% doesn't make it true. It's a false assumption, and therefore can't be used to derive accurate probabilities. – Arcanist Lupus Jun 16 '20 at 17:32
• @parseltongue, no, it doesn't. In your stated scenario, (if the character was randomly chosen) then there is an equal chance of the chosen character being any of the three. Therefore, there is a 2/3 chance that the character has glasses, and a 1/3 chance the character does not. To put it another way, the odds are 33% A, 33% B, 33% C. A and B have glasses, so the odds are 67% glasses, and 33% not. – Arcanist Lupus Jun 16 '20 at 19:16

EDIT: We were not aware of the rule that state if you guess incorrectly it means you instantly lose. That definitely makes it clear which you should do. We will definitely try to follow that going forward, thanks all!

# Guessing wrong lose the game: take a hint.

1. guessing has 1/3 to win and 2/3 to lose.
2. single hint: 1/3 you left with 1 charecter -> win at turn 2.
2/3 left with 2 characters -> 50/50 for a win. So 1/3+2/3*1/2=2/3 for wining.
3. Two hints: 100% win. 1/3 at turn 2, 2/3 at turn 3.

# If guessing wrong does not lose you the game: always take a guess.

Whether you just care about getting minimal questions on average (on expectation) or you care about getting fewer questions than your opponent. - In both scenarios taking a guess is better.

Comparison of the number of turns on expectation:

## If you guess:

1. got it right the first time with probability 1/3
2. got it right the second time with probability 1/3
3. got it right the third time with probability 1/3

Thus the expected number of turns is two. E(turns)=1/3*1+1/3*2+1/3*3=2

## If you go with a hint

1. 1/3 you leave 1 left, then win on the next turn.
2. 2/3 you leave 2 left, then either win on the next turn or on the 3rd turn.

So the calculation is E(turns)=1/3*2+2/3*1/2*2+2/3*1/2*3=2.333

By comparing expected turns we get that taking guesses will find the right character faster.

• As above this is incorrect. A wrong guess is instant loss. – StartPlayer Jun 16 '20 at 7:17
• @ArcanistLupus, you right, tnx, fix that. – Cohensius Jun 17 '20 at 5:27