# Return from Hiatus and Plants vs. Zombies

*By Steve On January 12, 2012 · 2 Comments*

Sorry for the delays, guys. I know all seven of you are **dying** for more lessons. I have to once again apologize, though, as this is going to be a lame entry about video games and math. I’ve just returned from being out of town for the past month, and have been having a little fun with the latest *Plants vs. Zombies* update, but, being a bit of a statistics nerd, I was particularly interested in the bronze, silver, and gold mystery sprouts you can buy for your zen garden. The description states that these new sprouts offer a 25%, 50%, and 100% chance of a unique plant, and cost $15k, $30k, and $50k, respectively. Now, at first glance, I thought this was pretty pointless. If you’re buying 4 bronze sprouts or 2 silver sprouts per unique, you’re just spending $60k average per unique vs. the ensured $50k for the gold. But 75% chance of getting a duplicate for the bronze? What’s the point? If you have less than 30 of the 39 total plants, then you’d have better odds with the regular mystery sprout. I shrugged it off as a gimmick.

A day or so later, I reflected back on the probability structure of the mystery sprouts. The descriptions were vague enough â€“ perhaps I misinterpreted it? Perhaps the 75% chance of failure for bronze means it reverts to a standard mystery sprout, rather than a guaranteed duplicate? If so, then the probabilities of unique for the bronze and silver are actually higher than 25% and 50%, though it scales down the more unique plants you have. However, this still means there’s thresholds, which means there’s fun to be had. Let’s make some charts!

The charts above outline the possible choice paths for each mystery sprout type, as well as their corresponding (accumulated) probabilities, leading up to the final probability formulas for each outcome. However, this in itself isn’t too useful to us. What we *really* want is a cost/probability relationship, to figure out which purchase is the most cost-effective. The graphs below show the average cost (in thousands) to the number of unique plants currently in your zen garden. (I didn’t bother rendering the graph for gold, as it would be a solid horizontal line at the 50 mark, since the probability is always 100%.)

Now we’ve got some interesting results. Here’s what we can infer from these two graphs:

- Regular is more cost-effective than bronze until you’ve reached 31 plants.
- Bronze is more cost-effective than gold until you’ve reached 36 plants.
- Silver is never cost-effective.

Anyway, that’s it for my nerdgasm. Was it as good for you as it was for me?

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Brilliant; I felt like the only person who was troubled by this! I think your interpretation is the most likely to be correct but have you also considered that the special sprouts and the regular sprouts might be in reversed positions in your diagrams (i.e. the special sprout’s effect only kicks in if random chance fails to produce a unique plant)? Would that change things very much and possibly give the silver sprout a window of cost-effectiveness after all?

@Simon: That would be an interesting consideration, but after looking into it, I think ultimately the probability will be the same. In both scenarios, the path kicks out of the chart as soon as the unique condition is met, and since the total probability of the unique is the sum of hitting the unique on the first try vs hitting it on the second try, the final formula will remain unchanged. So, nope! Silver is still worthless!