Modeling a promotion with balls and urns

In summary, the site where the player plays online offers a promotion in which the odds of getting a ball into a specific urn is 1 in 350. To win the promotion, the player must spend an average of 3500 points per shot.
  • #1
moscafly
2
0
Hi, I'm an online poker player and i need some help modeling a promotion that the site where i play offers. I think it's rigged and intend to prove it but I'm short on skillz.

The promotion is described here and i already made a model but can't solve it. It goes like this:

Supose you have 72 distinguishable urns. every one of them exactly alike except for 18 of them, which are smaller. let's call them "rare" urns.

A gun randomly fires undistinguishable balls towards this urns, and each ball thrown goes always to one of the urns (no missed shots). The urns allow infinite balls to fall on them (they don't get full)

Suppose the odds of throwing the ball to one of the 18 smaller urns is 1 in 350. Let's say urns 1 thru 18 are the "rare" ones.

let r= 1/350 (r stands for "rare" urn)

then the probabilty "p" of the gun throwing to a regular urn must be so that

18r + 54p = 1 solving for "p" it stands that the odds of getting the ball to a normal urn is roughly 1 in 56.9

Now suppose firing the gun costs 10 points.

The question goes like this: How many points on average do you have to spend in order to achieve a 95% probability of getting at least 1 ball in every urn?

Poker site claims that the last winner spent 54,000 points and manage to hit every urn. I call horse***.

Of course, they don't give away the exact odds of hitting a "rare" urn and they might be different kinds of "rares", but from a very small sample in a poker forum, people seem to be hitting "rare" urns spending an average of 3500 points, hence i estimated the odds of getting a ball into one of this urns to be 1 in 350.

please help, thanks in advance!
 
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  • #2
If it takes you on average 350 shots to get into one of the 18 urns, then the probability of getting into a specific urn is 1/(18*350).

In such a case the expected number of shots to fill all the rare urns (and hence all the urns at that point I'm going to assume) is

350+(18/17)*350+(18/16)*350+...+(18/1)*350

You can solve this by noting that once you've gotten a ball into k urns, the probability of getting another urn filled on your next shot is (18-k)/18*350, so it takes on average 18/(18-k)*350 shots to fill the k+1st urn. So the expected number of points spent is about 220000
 
  • #3
Thanks!

i see you didn't include the 95% assuarance... did you managed to calculate the EV(total points spended until last urn is filled)?
or is it more like a rough estimate?
 

1. How does the balls and urns model work?

The balls and urns model is a mathematical model used to represent the probability of different outcomes in a promotion. In this model, the balls represent the individual items or prizes being offered and the urns represent the different levels or tiers of the promotion. The number of balls and urns can vary depending on the specifics of the promotion, but the model is based on the concept of randomly selecting balls from urns to determine the likelihood of winning a particular prize.

2. How is the probability of winning calculated in this model?

The probability of winning in the balls and urns model is calculated by using the concept of combinations. The formula for calculating combinations is nCr = n!/(r!(n-r)!), where n is the total number of items or balls and r is the number of items being selected. For example, if there are 10 balls in the urn and 2 are being selected, the probability of winning would be 2/10 = 1/5 = 0.2 or 20%.

3. Can the balls and urns model be applied to different types of promotions?

Yes, the balls and urns model can be applied to a variety of promotions, such as raffles, sweepstakes, and contests. The key is to identify the number of items or prizes being offered and the number of levels or tiers in the promotion, and then use the appropriate combination formula to calculate the probability of winning.

4. Are there any limitations to using the balls and urns model for promotions?

While the balls and urns model can provide a useful representation of the probability of winning in a promotion, it does have some limitations. For example, it assumes that all balls are identical and that each ball has an equal chance of being selected. In reality, this may not always be the case, as some prizes may be more desirable or have higher odds of being won.

5. How can the balls and urns model be used to optimize a promotion?

The balls and urns model can be used to optimize a promotion by adjusting the number of balls and urns to achieve a desired probability of winning. For example, if a company wants to increase the chances of winning for a particular prize, they can add more balls to that urn or reduce the number of balls in the other urns. This can help attract more participants and create a more appealing promotion.

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