Problem Setup

You start with 1 amoeba. Each minute, an amoeba independently:

  • Dies with probability \( \frac{1}{4} \)
  • Does nothing (\( \to \) 1 amoeba) with probability \( \frac{1}{4} \)
  • Splits into 2 amoebas with probability \( \frac{1}{4} \)
  • Splits into 3 amoebas with probability \( \frac{1}{4} \)

Each new amoeba behaves identically and independently.

What is the probability that the entire population eventually dies out?

Step 1: Model as a Galton–Watson Process

Let \( X \) be the number of children an amoeba produces (including 0 if it dies). The offspring distribution is:

\[ P(X = 0) = \frac{1}{4},\quad P(X = 1) = \frac{1}{4},\quad P(X = 2) = \frac{1}{4},\quad P(X = 3) = \frac{1}{4} \]

Step 2: Compute the Expected Number of Offspring

\[ \mathbb{E}[X] = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{4} + 2 \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} = \frac{0 + 1 + 2 + 3}{4} = \frac{6}{4} = \boxed{1.5} \]

Since the expected number of offspring is greater than 1, extinction is not guaranteed, but it is possible.

Step 3: Use the Extinction Probability Equation

Let \( q \) be the probability of eventual extinction. For Galton–Watson processes, \( q \) is the smallest fixed point in \( [0,1] \) of the generating function:

\[ f(s) = \mathbb{E}[s^X] = \frac{1}{4}(s^0 + s^1 + s^2 + s^3) = \frac{1}{4}(1 + s + s^2 + s^3) \]

We solve:

\[ q = f(q) = \frac{1 + q + q^2 + q^3}{4} \]

Multiply both sides:

\[ 4q = 1 + q + q^2 + q^3 \Rightarrow q^3 + q^2 - 3q + 1 = 0 \]

Step 4: Solve the Cubic Equation

\[ q^3 + q^2 - 3q + 1 = 0 \]

Check rational roots (Rational Root Theorem): try \( q = 1 \):

\[ 1 + 1 - 3 + 1 = 0 \Rightarrow \boxed{q = 1} \text{ is a root.} \]

Factor:

\[ (q - 1)(q^2 + 2q - 1) = 0 \]

Solve the quadratic:

\[ q = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2} \]

Only root in \( [0,1] \) besides 1 is:

\[ q = -1 + \sqrt{2} \approx 0.4142 \]

Final Answer

\[ \boxed{ \text{The probability that the amoebas eventually die out is } \boxed{-1 + \sqrt{2}} \approx \boxed{0.4142} } \]

Reference