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  #931  
Old 11-25-2014, 02:35 AM
VirtualBoyFreak VirtualBoyFreak is offline
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iPhone 4S, iOS 7.x
 
Join Date: Apr 2011
Location: Spain
Posts: 2,183
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Quote:
Originally Posted by cloudpuff View Post
Gahhhhhh, I deleted it yesterday to make room for the Monument Valley update. Off to redload lol.
Today's update drastically reduces installed size: from 317MB to approx. 76MB. Finally able to keep it in my device!
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  #932  
Old Yesterday, 02:09 AM
BenW BenW is offline
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iPad mini 2, iOS 7.x
 
Join Date: Feb 2014
Posts: 73
Default Bonus Card Distribution

It occurred to me that the new mechanism for displaying bonus cards (always showing three choices at the top) is likely to cause some highly counterintuitive behavior. For instance, suppose the high card on the board is 384. If the bonus card is chosen uniformly from [6, 12, 24, 48], then seeing [12, 24, 48] on the screen actually implies a GREATER than 1/3 probability of the bonus card being a 48! The logic is as follows, again assuming high card 384:

Bonus card 6 (25% of the time): screen must show [6, 12, 24]
Bonus card 12 (25%): screen shows [6, 12, 24] or [12, 24, 48] with equal probability
Bonus card 24 (25%): screen shows [6, 12, 24] or [12, 24, 48] with equal probability
Bonus card 48 (25%): screen must show [12, 24, 48].

So there are six possibilities:
25%: Bonus card 6, screen shows [6, 12, 24]
12.5%: Bonus card 12, screen shows [6, 12, 24]
12.5%: Bonus card 24, screen shows [6, 12, 24]
12.5%: Bonus card 12, screen shows [12, 24, 48]
12.5%, Bonus card 24, screen shows [12, 24, 48]
25%: Bonus card 48, screen shows [12, 24, 48]

So now let's say you're playing the game, and the screen shows [12, 24, 48]. By the above table, there is actually a 50% chance that the bonus card will be 48, a 25% chance that it will be a 24, and 25% that it will be a 12! Likewise, if the screen shows [6, 12, 24], there is a 50% chance the the bonus card will be a 6, and 25% each for 12 and 24.

Empirical data: I just played a game and got five bonus cards while the high card was 384. In 4 out of 5 cases, the bonus card was the extreme value. (e.g. [6, 12, 24] -> 6, [12, 24, 48] -> 48). If the probability were 1/3 of getting the extreme card, as you'd naively expect, then the chance of getting 4/5 is less than 5%. But if the probability of getting the extreme card is 50%, as calculated above, then the chance of getting 4/5 rises to 37.5%.

This gets even wonkier when the high card goes higher. Take the case of the high card being 768. Then the game first chooses the bonus card uniformly from [6, 12, 24, 48, 96]:

1/5: bonus card 6, screen must show [6, 12, 24]
1/5: bonus card 12, screen shows [6, 12, 24] or [12, 24, 48]
1/5: bonus card 24, screen shows [6, 12, 24], [12, 24, 48], or [24, 48, 96]
1/5: bonus card 48, screen shows [12, 24, 48] or [24, 48, 96]
1/5: bonus card 96, screen must show [24, 48, 96]

Let's assume that the cases split evenly; e.g. if the high card is 12, the screen shows [6, 12, 24] or [12, 24, 48] with equal probability. Then there are 9 distinct cases:

1/5: bonus card 6, screen shows [6, 12, 24]
1/10: bonus card 12, screen shows [6, 12, 24]
1/10: bonus card 12, screen shows [12, 24, 48]
1/15: bonus card 24, screen shows [6, 12, 24]
1/15: bonus card 24, screen shows [12, 24, 48]
1/15: bonus card 24, screen shows [24, 48, 96]
1/10: bonus card 48, screen shows [12, 24, 48]
1/10: bonus card 48, screen shows [24, 48, 96]
1/5: bonus card 96, screen shows [24, 48, 96]

In this case, the choices shown on the screen yield the following probabilities for each bonus card:

[6, 12, 24] -> 6 (54.5%) 12 (27.3%), 24 (18.2%)
[12, 24, 48] -> 12 (37.5%), 24 (25%), 48 (37.5%)
[24, 48, 96] -> 24 (18.2%), 48 (27.3%), 96 (54.5%)

Empirical data: out of 7 times in a recent game where I got [6, 12, 24] or [24, 48, 96] while the high card was 768, the actual bonus value was the extreme value (6 or 96) three times, or 42.8%. Too small a sample size to mean much, but I'll continue to gather data.

Another expected consequence is that the middle ranges will be shown somewhat less often than the extremes. For instance, with the high card 768, you can expect that the bonus range will be shown as [6, 12, 24] 36.6% of the time, [12, 24, 48] 26.7% of the time, and [24, 48, 96] 36.6% of the time.

It would be interesting to gather statistics over a number of games and see if these predictions bear out. From the statistics we can probably reverse-engineer the logic the game uses as far as choosing bonus values and ranges.
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