# Talk:Bryophyta's staff

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## calculator in article

the calculated savings of the article assumes that otherwise you would be using natures and fire runes on each cast. if we understand that the other alching option is using a staff that saves fire runes, 75x the cost of fire runes should be subtracted from the savings of the staff in casting high alch. this makes the staff of fire currently a better option for high alching, though this article implies otherwise. 72.199.173.97 21:08, 6 September 2020 (UTC) -OSRSandChiII, 9/6/2020

You can use both Bryophyta's staff and a tome of fire at the same time, so it is possible to save both the fire runes and 1/15 of the nature runes. The saved cost is per cast, not for 15 casts. So the average savings per cast with a fire staff would have to be 5 times the price of a fire rune. But I think it would indeed be nice to add this as clarification. --Germaan (talk) 23:42, 6 September 2020 (UTC)

## Some calculations to clarify values

Write $X_{i}{\stackrel {\textrm {iid}}{\sim }}\,\mathrm {Bernoulli} \left({\frac {14}{15}}\right)$ for cast $i$ to be 1 if a nature rune is consumed and 0 otherwise. Then notice that $C_{T}:=\sum _{i=1}^{T}X_{i}\sim {\textrm {Binomial}}\left(T,{\frac {14}{15}}\right)$ is the number of nature runes consumed in $T$ casts. Further, notice that $S_{M}:=-M+\min _{C_{T}=M}T\sim \,{\textrm {NegativeBinomial}}\left(M,{\frac {1}{15}}\right)$ is the number of nature runes saved by the time there are $M$ nature runes consumed. With these random variables and their distributions, we can readily write down various statements about the staff in terms of expected values.

### Number of runes saved per cast

The number of nature runes consumed per cast is $X_{i}$ , making the number of nature runes saved per cast equal to $1-X_{i}$ . This has mean $\mathbb {E} \left[1-X_{i}\right]={\frac {1}{15}}$ . Multiplying this by the cost of nature runes gives the average savings per cast.

### Number of runes saved per nature rune consumed

For each nature rune consumed, there are $S_{1}$ runes saved. This has mean $\mathbb {E} \left[S_{1}\right]={\frac {{\frac {1}{15}}\cdot 1}{1-{\frac {1}{15}}}}={\frac {1}{14}}$ . Multiplying this by the cost of nature runes gives the average savings per rune consumed.

### Effective number of nature runes

With $M=1000$ runes stored in the staff, there will be $M+S_{M}$ nature runes spent before the staff depletes. This has mean $\mathbb {E} \left[M+S_{M}\right]=M+{\frac {{\frac {1}{15}}M}{1-{\frac {1}{15}}}}={\frac {15}{14}}M$ .

### Number of casts needed to "pay off" staff

The number of casts needed to pay off the staff is the smallest number $T$ so that $\mathbb {E} \left[\sum _{i=1}^{T}(1-X_{i})P_{\textrm {nature}}\right]\geq P_{\textrm {staff}}$ where $P_{\textrm {thing}}$ is the price of ${\textrm {thing}}$ . Simplifying, this expression reads that $T{\frac {1}{15}}P_{\textrm {nature}}\geq P_{\textrm {staff}}$ , or that $T\geq 15{\frac {P_{\textrm {staff}}}{P_{\textrm {nature}}}}$ . Therefore the number of casts needed to pay of the staff is simply $\left\lceil 15{\frac {P_{\textrm {staff}}}{P_{\textrm {nature}}}}\right\rceil$ .