A ball game is a state where you have $n$ white balls and $m$ black balls. The rule is that you remove first one ball from the cup. And without returning the first ball, you pick another.
$P(A)$ is defined as the probability of drawing two balls with opposite colors
$P(B)$ is defined as the probability of drawing two balls with the same color
A game is defined as fair if $P(A)=P(B)$.
A game is fair if and only if $n$ and $m$ are consecutive triangular numbers.
$(n,m) = (1,3) \ , \ (3,6) \ , \ (6,10) \ \ldots$
Image a cup with 2 white balls and 2 black balls. If you draw two balls with the same color, you win. If on the other hand you draw two balls with opposite colors, I win. Note that once a ball is drawn the ball is gone.
Is this a fair game? No, ofcourse not. After you pull a ball from the cup there are two of my colors, and only one of yours. Giving me a $P = 2/3$ chance of winning. This can be illustrated in the following diagram
Following a doted line means I win. Following a whole line, you win. There are more doted lines, than whole; hence I win. One can make this game fair by changing the balls
just count the lines, or do the simple math. Now a fun generalization is to find all configurations that allow a fair game. Suprisingly this is always two consecutive triangular numbers. I want to explain this to my class in an intuitive way, perhaps let them explore it.
- Is there a way to use the diagrams or else to obtain a intuitive explenation why the solutions are always two consecutive triangular numbers?