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I have 10 red balls, 10 blue balls, and 2 baskets.
You are free to choose the numbers of red balls, and the numbers of blue balls to put to the first basket. Then you have to put rest of the balls to the 2nd basket.
Then I will choose a basket and take a ball blindly.
If I get a red ball I will give you 2 dollars, but If I get blue ball you have to give me 3 dollars.
Will you accept my challenge? Why?
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I will split the balls like this
1 red in the first basket, 9 red and 10 blue in the second. My winning chance for each basket is then $100\%$ and $\frac{9}{19}\approx 47\%$
Assuming you choose a basket with a 50/50 chance.
My overall winning chance is $\frac{14}{19}\approx 74\%$
My expected winning is then $2\cdot\frac{14}{19}-3\cdot\frac{5}{19}=\frac{13}{19}\approx0.68\$$
So I will
Accept you challenge
If you can see the distribution of ball and choose the basket which is best for you, the I will
Not accept your challenge
With an equal amount of red and blue ball we cannot do better than $50\%$. Increasing the chance one basket will decrease the other. Since I can lose 3\$ but only win 2\$, I am expecting to lose money.
