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This is a probability question that I came up with, and have noticed some things that do not seem to be right. Here's a description of the question:

Suppose we have a fruit tree growing in our garden, which grows from scratch in steps (stages). At each stage, there are three equally likely components of the tree that could appear: a single branch, a double branch, or a fruit. A branch's growth is independent, and any one branch will stop growing only if a fruit appears in its last stage of growth. A double branch means that one branch turns into two branches. A single branch is sort of redundant, as in it only delays the growth of a fruit, which is not of our concern.

Denote the probability that we end up with exactly $n$ fruits by $f(n)$. What is $f(n)$?

I did derive a formula for $f(n)$, but I feel it is wrong. I first noticed that $f(0)=0$ as the tree won't stop growing until there is at least one fruit. Some initial values of $f(n)$: $$f(1)= \sum_{i=0}^{\infty} P(i \ \text{single branches and one fruit})=\frac 13+\frac 13\cdot\frac 13+\frac 13\cdot\frac 13\cdot\frac 13 \cdots =\frac 12 $$ By symmetry, the probability of occurrence of a double branch after however many number of single branches will also be $\frac 12$.

$$f(2)= P(\text{double branch})\cdot [P(\text{fruit on one of the branches of the double branch})]^2\\=\frac 12\cdot \left(\frac 12\right)^2 =\frac 18$$

For the general case, the following recursion is obtained: $$f(n)= P(\text{double branch})\cdot P(\text{fruit})\cdot f(n-1) \\ \implies f(n)=\frac{1}{2^{2n-1}}$$

The problem is, when I sum $f(n)$ I'm not getting the expected result: $1$. $$\sum_{n=1}^{\infty}f(n)= 2\sum_{n=1}^{\infty}4^{-n} =\frac 23\overset{?}{\ne} 1$$ What is happening here? Any pointers to where I'm wrong will be highly appreciated. Thanks.

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  • $\begingroup$ @PeterForeman: That the expectation is infinite does not imply a non-zero probability for infinitely many fruits (e.g. the one-dimensional simple symmetric random walk has infinite expected return time to the origin but return probability $1$); see my answer. $\endgroup$
    – joriki
    Jun 1, 2020 at 18:38

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Your recursion is wrong because you’re only accounting for the possibility that one particular branch bears $1$ fruit and the other bears $n-1$. This happens to yield the right result for $n=2$ because in that case $(1,1)$ is the only possible combination of fruit counts. But $n=3$ can be realized as $(1,2)$ and $(2,1)$, so the probability is twice what you calculate, and beginning with $n=4$ both branches might bear more than one fruit. The correct recursion is

$$ f(n)=P(\text{double branch})\sum_{k=1}^{n-1}f(k)f(n-k)\;. $$

Multiplying this by $P(\text{double branch})$ and defining $g(n)=P(\text{double branch})f(n)$ yields

$$ g(n)=\sum_{k=1}^{n-1}g(k)g(n-k)\;, $$

with $g(1)=\frac14$. Here are the first few values:

\begin{array}{c|c|c} n&g(n)&f(n)\\\hline 1&\frac14&\frac12\\ 2&\frac1{16}&\frac18\\ 3&\frac1{32}&\frac1{16}\\ 4&\frac5{256}&\frac5{128} \end{array}

Contrary to what has been stated in the comments, the probabilities should sum to $1$, since the average number of descendants of each branch is $1$ and thus the extinction probability for the branches is $1$ (see e.g. Wikipedia). The statement in the comment that the expected number of fruits is infinite is correct; this does not imply that the probability to obtain infinitely many fruits is non-zero. It would, however, be non-zero if you very slightly tweak the probabilities such that each branch generates more than one branch on average.

Edit:

Actually, the problem can be mapped to a simple symmetric random walk in one dimension. As you noted, you can ignore the single branches, since they’re just a delay. So the probability to eventually get a fruit or a double branch is $\frac12$ each. It doesn’t matter in which order we process the branches; all that matters is the number of active branches. So with probability $\frac12$ you decrement the number of active branches, and with probability $\frac12$ you increment it; so the number of active branches is a simple symmetric random walk starting at $x=1$. Each decrementing step produces a fruit, so the number of ways to produce $n$ fruit in $2n-1$ steps from $1$ to $0$ without hitting $0$ in between is the $(n-1)$-th Catalan number $C_{n-1}$, and the corresponding probability is $2^{-(2n-1)}C_{n-1}$, in agreement with the above table and recurrence relation.

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