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# Kinda stuck on this question

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A certain game is won when one player gets 3 points ahead. Two players A and B are playing the game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?

Attempt:

The green dot represents the score, and for every point A gains it goes down by one, and for every point B gains, it goes right by 1. Whenever it touches the blue line, A wins, and when it touches the red line, B wins, so I was thinking of using something similar to Catalan's formula?

I think my method seems way too complicated; if anyone else has a simpler solution feel free to write it.

Mar 25, 2022
edited by Melody  Mar 25, 2022

### 3+0 Answers

#1
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A certain game is won when one player gets 3 points ahead. Two players A and B are playing the game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?

We do not care about the number of players we only care about the number of points that A is in front.

So draw a number line.

It goes from -3 to 3   (a total of 6 units)

If the difference gets to -3 then A loses, if it gets to +3 A wins.

A starts at +1

A must go a 2 units to the right to win and 4 units to the left to lose.

So the liklihood of a win is 4/6   and the likelihood of a loss is 2/6

So he has a 2/3 chance of winning.

This is not a formal solution.  I don't know how to solve fomally but I believe this is effectively a speed test so intuition is of utmost importance.

Mar 25, 2022
edited by Melody  Mar 25, 2022
#2
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Thanks melody! I suspected that the solution would be simpler than I thought.

textot  Mar 26, 2022
#3
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You are very welcome.

Melody  Mar 26, 2022