Daily Math Puzzle: 2026-04-13

100% — Let P be the probability that the entire lineage eventually dies out. We need to consider the fate of the initial single organism: 1. **With 50% probability, the initial organism dies.** In this scenario, the lineage immediately dies out. So, this contributes 0.5 * 1 to the total probability P. 2. **With 50% probability, the initial organism divides into two new, independent organisms.** For the *entire original lineage* to eventually die out in this case, *both* of these new organisms' lineages must also eventually die out. Since each new organism behaves identically and independently, the probability that one of their lineages dies out is P. Therefore, the probability that *both* new lineages die out is P multiplied by P (or P^2). Combining these two possibilities, we can form an equation for P: P = (0.5 * 1) + (0.5 * P^2) P = 0.5 + 0.5P^2 Now, we solve this quadratic equation for P: Multiply by 2 to clear the fraction: 2P = 1 + P^2 Rearrange into standard quadratic form: P^2 - 2P + 1 = 0 This is a perfect square trinomial: (P - 1)^2 = 0 Taking the square root of both sides: P - 1 = 0 P = 1 So, the probability that the entire lineage will eventually die out is 1, or 100%. This is often counter-intuitive, as it seems there's always a chance for the population to grow, but for the lineage to *never* die out, it must grow infinitely large, which requires an unbroken chain of divisions without any organism dying, which becomes infinitesimally improbable over infinite time.