Lessons in Probability from the Serengeti
A lioness lies flat in the dry grass, eyes locked on a herd of zebras in front of her. She has not eaten in three days. One drifts towards her, head down, chewing. Her shoulder blades rise and fall as he inches forward. Twelve metres. Eight. Six. She gathers her hind legs beneath her. The zebra lifts its head. For a fraction of a second nothing moves. Then the herd erupts in a wall of dust and hooves and she is left alone in the flattened grass.
She circles downwind for an hour. A wildebeest calf has wandered from its mother. She moves with the patience and laser focus. The calf, oblivious to her, turns the wrong way and she breaks into a sprint. But the mother comes out of nowhere, horns low and the calf is gone behind a wall of muscle before her paws can close.
She tries again. A gazelle, alone, distracted, perfect. She closes the distance in silence, but then suddenly the wind shifts and the gazelle immediately picks up her scent.
The fourth time, an impala stands broadside, drinking, head low. She shifts her weight in the grass, to get ready to attack, but then a single twig snaps. The impala is already running.
The fifth time, she chooses a gazelle at the edge of the herd, picks her line, waits for the moment the head drops. The sprint is perfect. The angle is perfect. But the gazelle simply runs faster than she does and the gap between them widens with every stride until the chase ends itself.
Five hunts. Five failures. The sun has moved across the sky and she is hungrier than when she started. Is something wrong with her? Are her eyes weakening with age? Has she lost a step? Is she hunting the wrong prey, in the wrong place, at the wrong hour? Should she change her technique, try something different tomorrow?
No. A lioness succeeds in roughly one of every five hunts (Schaller, 1972). That is what her life looks like. Everything is running exactly as it always has and the kill, when it comes, will sit somewhere in the tail of a curve she is fully familiar with.
To us often such a streak feels like proof that something is broken. She will just continue and understands something intuitively that we with our big fancy brains are often blindsided by. I am aware of the anthropomorphism. A lioness most likely does not "understand" anything in the way we do. But the point of the image is precisely that she behaves as if she had grasped what we often struggle to.
The Bridge Between Probability and Time
What we are witnessing is a universal generator, which takes one input, p, the probability of success on any single attempt, and produces outputs.
For the lioness, p is around 0.2 (Schaller, 1972). For a cheetah, closer to 0.4 (Caro, 1994). For a polar bear stalking seals on the ice, less than 0.05 (Stirling, 1974).
If you watched one hundred hunting sessions of the lioness, she would still be hungry after five attempts in one of every three. Five failures in a row is not a bad day. It is what a 20 percent success rate looks like when you actually watch it unfold across a hunting session. P(fail) = 0.8. P(5 fails in a row) = 0.8⁵ ≈ 0.33 - so one in three hunting sessions begins with five failures.
The generator works the same way in every domain where outcomes are produced through repetition. Hunting, experimenting, investing, selling, dating … the list goes on. It just takes a static number and unfolds it through repetition. One attempt with probability p becomes a sequence of attempts and the sequence has a shape. After n tries, the chance of having succeeded at least once is 1 minus (1 minus p) to the power of n. The curve rises steeply at first, then bends, then crawls toward certainty without ever reaching it. You can run the machine forever and never be guaranteed an outcome. You can only push the probability closer to one.
The whole machinery rests on two assumptions. The first is that p stays still. In reality p moves, sometimes upward, sometimes downward as the conditions turn against you. The second assumption is that the attempts are independent, that nothing carries over from one to the next. In a real hunt, the lioness tires. Her scent lingers in the grass. The herd grows wary. In a human life, you learn and your reputation precedes you. So the generator is a clean model of a dirty world.
The size of the attempt budget
Once you accept a generator, the central question of any project is no longer "will it work" but "how many shots do I have, until I run out of energy / money / … .". Each failed hunt costs calories. A successful one gives many back, but it needs to happen before the lioness loses the strength to go on the one hunt where she makes a killing. It is present in the Serengeti but also in finance, as each rejected investment is bounded at minus one hundred percent, but a successful one can be a hundred times that.
The arithmetic is the same, it only wears different names. A founder has a runway measured in months. The number is just a bank balance divided by a burn rate. Each month takes one of them, whether or not anything got built. A graduate has two more application cycles and three professors still willing to write for her. Those letters do not stay strong forever. Two cycles from now the same year stops reading as a pause and starts reading as a verdict. Each attempt draws down something specific and finite. Each kill, when it comes, returns more than it cost. The question is never whether the next attempt will succeed, but whether the budget lasts long enough for one to arrive.
The shape of the payoff distribution determines how aggressively you should sample, and how much variance you can tolerate. A symmetric payoff does not justify a thick stack of misses. The budget runs down at the same rate but nothing in the run pays back enough to climb out of the hole. A long thin tail of upside on the right does justify them; the misses are simply the way through to it.
Not every failure simply just subtracts a slice. Some end the run. A lioness who breaks her leg on attempt three has no attempt four. A founder who personally guarantees the wrong loan does not get a second venture in the same decade. These are absorbing states (Peters, 2019), the moves you make once and only once. The simulation above pauses the instant the budget reaches zero and it does so because in the real Serengeti there is no continue button.
The Machine You Stand In Front Of
The Lioness was born in the savanna and has a very limited choice about here geographical location. But you do.
Two engineers leave the same university in the same year with roughly the same talent. One arrives in San Francisco in 2011 almost by accident, takes a job at a small company nobody has heard of and ten years later owns a house he never would have dared to dream of. The other joins a more prestigious firm in Frankfurt, works harder, sleeps less and is, after ten years, promoted, but 10x under the lifestyle of his college in california. The variance between their lives was decided the day they walked into different rooms. Everything since has been furniture.
Or, two friends are single in their late twenties in the same city. Both kind and ready. One of them is conventionally attractive in the way that gets noticed across a room. The first goes on perhaps eighty first dates over three years. The second goes on twenty. Both bring the same care to each one. Both listen the same way. The first ends up with twenty second dates, six real connections and the quiet luxury of choosing between them. The second ends up with two second dates, one real connection and a partner he is grateful for. They are both with someone, the first had choice the second a chance.
The generator most often does more of the work than the attempts do (Pluchino et al., 2018). Same hours, same care, same talent, but different outputs by an order of magnitude. We have been taught since childhood that effort is the lever. Effort is a lever, but a small one compared to the underlying generator. You can pour your heart, soul and last squeeze of your life energy into the wrong one and it will not care.
When did you last actually look at the machine you are standing in front of? What would you see if you drew it long enough? What p would you need to see to live the generator? But leaving a generator most often does not come cheap. Walking out of a city means losing friendships to a certain degree. Or leaving a field is forfeiting years of compounded network, competence and reputation. This question is just so essential as inside the right generator, mediocre effort can produce good outcomes, but inside the wrong one, even brilliant effort might produce nothing at all. The truly generator-aware person spends less time optimising individual attempts and more time choosing the machine they are willing to feed for a long time.
Process beats outcome
If p is the probability that any given attempt succeeds, then a single failure tells you almost nothing about whether your process is sound. Neither do two or three and more in a row. Someone who understands the generator stops judging themselves by individual results and starts judging themselves by whether they are running the right machine and how often they can do it. This is the antidote to outcome bias, a very expensive error in decision making (Baron & Hershey, 1988).
The sun sets on the Serengeti. The lioness has not eaten today. She lies down in the grass and her flanks heave once, then settle. Tomorrow she will hunt again.
References
Baron, J., & Hershey, J. C. (1988). Outcome bias in decision evaluation. Journal of Personality and Social Psychology, 54(4), 569–579.
Caro, T. M. (1994). Cheetahs of the Serengeti plains: Group living in an asocial species. University of Chicago Press.
Peters, O. (2019). The ergodicity problem in economics. Nature Physics, 15(12), 1216–1221.
Pluchino, A., Biondo, A. E., & Rapisarda, A. (2018). Talent versus luck: The role of randomness in success and failure. Advances in Complex Systems, 21(03–04), 1850014.
Schaller, G. B. (1972). The Serengeti lion: A study of predator-prey relations. University of Chicago Press.
Stirling, I. (1974). Midsummer observations on the behavior of wild polar bears (Ursus maritimus). Canadian Journal of Zoology, 52(9), 1191–1198.