### Brain Teasers

# Balls in a Jar

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

You're walking down a street, when you see some people gambling on the roadside.

The dealer says:

"There are 10 black balls and 10 white balls in this jar. You are blindfolded and you randomly pick 10 balls. If 5 of the balls you picked are black and the other 5 are white, you lose $2. Otherwise you win $1."

Is the gamble fair?

The dealer says:

"There are 10 black balls and 10 white balls in this jar. You are blindfolded and you randomly pick 10 balls. If 5 of the balls you picked are black and the other 5 are white, you lose $2. Otherwise you win $1."

Is the gamble fair?

### Hint

You know that, in most gambles, the dealer wins more than he loses. If not, why does he hold the gamble?### Answer

The number of combinations to take 10 balls out of 20 is 20!/(10!)^2=184756.The number of combinations to take 5 balls of each color is (10!/(5!)^2)^2=63504.

This means in 184756 games you lose 63504 games and win 121252 games in average, totaling a loss of $5756. In other words, you lose about $0.03116 per game in average.

Therefore, the gamble is not fair.

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## Comments

Very well done, I was going in a different direction in a sense, liked it a lot and keep these coming

good one. i didnt get it (i just guessed, hey 50/50 chance rite?) but had fun doin it.

i thought u had 50% of a chance oh well im dumb and i got 2 get used 2 it

Guess I was just too tired to do the math on this one, although I know it wouldn't make sense for a gamble to set himself up for a loss, so I guessed!!! great one though may have to try it on someone myself although with my luck I would probably lose the first 10

That is a very suggestive title

Had the answer. I was glad you told me how it worked mathematically.

i feel dumb i thought well i dont know what i thought

ahhh you got me

very witty and a clever teaser thanks

I loved this so much going to have nightmares about my probability class taken from the infamous Robert Waller!

Keep em coming.

cariapat

Keep em coming.

cariapat

that was so easy!!!!

I don't think most gambling casinos pay out much more than 97 cents on the dollar, and perhaps much less. I'd say the gambler was fair, as he ought "earn" something for his work. Anyway, good calculations!

The math is good but you don't even have to worry about the formula. You are going to pick 10 balls out of the 20. You have an equal chance that you come back with (0 black, 10 white) or (1 black, 9 white) or (2 black, 8 white), etc. Out of these 20 possibilities, 2 are (5b,5w) or (5w,5b). So 1 in 10 is 5b,5w while 9 in 10 is something else. So you lose $2 10% of the time and make $1 90% of the time. Take the bet, if you play 100 times you'll make $70.

I screwed up in the previous post: use what I said to understand the mechnism of unfairness but my math is incorrect (oiverly simplistic, sorry). The math in the original answer is correct.

Looks like I have to back to my textbooks and learn all about permutations and combinations before I do any more of these probability problems

that was easy and fun.i got it.it's not fair.am i right?lol of course!

Cool one!

I don't like problems where I have to use a calculator...

Nevertheless, the approach to this problem is very basic if you're comfortable with counting.

Nevertheless, the approach to this problem is very basic if you're comfortable with counting.

A bit confused again... why are we squaring the combination for 5 balls of one color out of 10. If 10!/(5!)^2 is the number of combinations of picking 5 black balls... by default isn't this the same set of combinations for 5 white since you are either picking a white or a black? I manage to get most of this questions with a bit of thought but this one seems to elude me. Thanks

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