The sixth in a series of seven fables/lessons/meditations on probability.
Another day of hiking brought the teacher and the student to an empty hut by a mountain stream. “We will rest here a while, and wash our clothes,” the teacher said.
When they had laid their clean clothes on sunny rocks to dry, the student pointed to the clouds gathering in the valley below. “Looks like rain. Should we be worried?”
“The rains have reached this place only once in the last 100 years,” the teacher said. “What is the probability that they will reach us today?”
The student thought for a moment. “Let’s say 30 storms reach the valley every year. Over 100 years, that’s 3,000 storms. But only one of them has reached the mountaintop here. So the probability is roughly 1 in 3000.”
“A very low probability,” the teacher nodded. Then she moved some of her half-dried clothes from the rocks to the inside of the hut. The student wondered why.
Later, the teacher plucked a blade of grass. “There is no dew,” she said. “This occurs less than once each month. What is the probability the rains reach us now?”
“Well, there’s no dew roughly 10 times per year,” the student said. “In the last century, that’s 1000 times. And at least 999 of those times, it didn’t rain. So the chances of rain are still at most 1 in 1000.” The teacher nodded, but moved more clothes inside nevertheless.
Later, a large brown bird flew directly overhead, barely 50 feet up. “The hawk-eagle flies low, just above the treetops,” the teacher said. “I glimpse such a sight only once a year. What is the probability the rain reaches us now?”
“Still small,” the student said. “The hawk-eagle flies low once a year, but it only rains once every century. So even if this is a sign of rain, the probability is still at most 1 in 100.” The teacher moved still more of her clothes inside the hut.
Later, as clouds began to darken the sky, the teacher pointed west, where pale stripes of color stood out against the gray sky. “A rainbow occurs in the West only once every 5 years,” the teacher said. “What is the probability that the rains reach us now?”
“Still low!” the student said. “There’s been a rainbow in the west 20 times in the last century. But it’s only rained one of those times. The probability is still just 1 in 20. There’s nothing to worry about.” Nevertheless, the teacher moved the last of her clothes inside the hut.
Just then, a light rain began to fall. The student scrambled to gather her clothes, but within minutes, the downpour was torrential, and everything was soaked through.
“I should have known better,” the student sighed from the shelter of the hut. “You tricked me again.”
“No trick,” the teacher said. “I told you everything.”
“How can that be?” the student said. “The odds should never have gotten higher than 1 in 20. What did you leave out? I used all three signs.”
“No.” The teacher shook her head. “You used one sign, then a different sign, then a third sign. You never used them all. How many days do you think all three signs have occurred?
The student blinked. “I don’t know.”
“To my knowledge,” the teacher said, “only twice: today, and once many decades ago, when I was a young girl.”
“And you still remember that day?”
“Of course,” the teacher said. “It was the day that rain reached this place.”
Further Thoughts
The focus here is conditional probability—the probability that one event will occur, given that another has already occurred.
In some sense, all probability is conditional. When we ask “What’s the chance of rain tomorrow?” we’re really asking, “What’s the chance of rain tomorrow, given the conditions today?” When we ask, “What’s the probability that the Patriots win the Super Bowl” we’re leaving out the rest of the question: “given their success so far this season?” To compute such probabilities, you’ve got to know what information to use, and how to use it.
The student in this parable deserves credit for applying the first key rule: Finding out new information may change your probability.
For example, say we’re rolling a standard die. What’s the probability we roll a 4? Simple: there are six possibilities, so it’s 1/6.
But what’s the probability we roll a 4 given that our roll is even? Now we’ve got new information, and it changes our outlook. We can toss out 1, 3, and 5 as possibilities. That leaves 2, 4, and 6. So there’s a 1/3 chance that we rolled a 4.
Or, try this: What’s the probability we roll a 4 given that our roll is a perfect square? Well, there are only two perfect squares on a die—1 and 4. So the probability is ½.
Finally: What’s the probability we roll a 4 given that our roll is even and a perfect square? There’s only one possibility fitting this description—4 itself. So the probability is 100%. It’s this final step that tripped up the student. She used one piece of information. Then she used the others. But she never used them all at once.
Get the pdf, or check out other stories in the series!
I’d like to thank my father, James Orlin, for providing some foundational ideas for these stories, as well as helpful feedback and conversations. Also for being one smooth operator.
However, suppose in another version of the story, the first sign was a robbery in the village below, the second sign was a shooting star exactly at midnight, and the third sign was the birth of twins in the village. Even if all three signs co-occurred today and on the last day of the 100-year rain, it would be desperately bogus to suppose that there was a high probability of rain today. Signs in real-world examples have to have some sort of relationship with the event to be predicted, or they are mere superstitions.
Yeah, a very good point. I specifically picked three signs with a cause-and-effect relationship to rain.
But suppose this unfolded for a million years, and the confluence those three signs (shooting stars, robberies, and twins) continued to predict the rainstorm of the century, each and every time. It’d be harder to dismiss that as coincidence, and we might have to start looking for a causal framework whereby those superstitions could become science!
Even so. Continental drift was first proposed in 1912 (if not sooner) but always foundered on the rock (literally) of the lack of a physical explanation. How could continents move through the oceans? The development of plate tectonics told us the how, after which the what was easier to accept even though there was no more direct evidence than before.
Probability- the honing of uncertainty to measure the likelihood of occurences using the past as a guide……(just whimsy)
Whimsy perhaps, but pretty true as well!
(Often we use abstract reason as a guide, in addition to past experience.)
“The rains have reached this place only once in the last 100 years,” the teacher said. “What is the probability that they will reach us today?”
The student pretended to think, then replied, “Well this is a year, and so the probability must be one in one hundred.”
I like that. Some Alice-in-Wonderland logic, that is.
Reblogged this on Just another complex system.
I like what you are doing!
This fable illustrates two other concepts in the real-world application of probability:
1) Ultimately, we must act upon a probabilistic event as though it is not probabilistic. The chance of rain may be 30%, but we cannot carry 30% of an umbrella–we must either choose to bring the umbrella, as though the probability is 100%, or leave it at home, as though the probability is 0%.
2) The response to a probabilistic event is weighed against the possible consequences. In the case of this fable, the consequence of guessing wrongly that no rain would fall is wet clothes–hardly the worst imaginable fate. If the consequence were, say, being eaten by a tiger, the student would be more inclined to play it safe. Similarly, my chance of being in a violent crash are pretty small each time I drive my car; I’ve driven many thousands of miles in my life, and only been in one injury collision (and I was not the one injured). Yet, I wear a seat belt every time, as though the chance of being in a collision is 100%, because the consequences can be dire, and wearing a seatbelt is easy.
The difficulty with acting upon a probability is that the options may each have good and bad components to the outcomes. Do you invest in that startup company? Do you ask the girl out? Do you become a Catholic, Buddhist, or atheist?
Love the fables! I will be sure to add them to the Monty Hall riddles and Prisoner Dilemmae in my repertoire. Thanks for the thinks!
How much rain would have had to have fallen and how big of a basin would there have to be for a stream to still have water after many decades without rain?
Apparently the “mountain hut” is downstream of some higher terrain on which the rains have fallen more recently; the puzzle only needs the neighbourhood of the hut to have been rain-free, aside from one day, for a century.
However:
“Another day of hiking brought the teacher and the student to an empty hut by a mountain stream.”
I infer that it is evening, so the sun is in the West.
If there were a rainbow, it would thus be in the east; if you are looking at a rainbow, the light-source causing it is behind you.
“A rainbow occurs in the West only once every 5 years,” the teacher said, neglecting to mention that this only ever happens in the morning.
The rest of this series is great, but I think this one has two major flaws:
1) In my opinion there’s a more important conditional clause here. What’s the chance of rain given that clouds have gathered and it *looks* like rain? Likely much more likely than the average day (although of course it depends on an unknown, namely how often it looks like rain. But you could still assign some prior to that based on your experience in other places).
2) Although the three named events may correlate with rain, there’s nothing in the story that suggests that the student should *believe* that they correlate with rain.
I think both problems could be fixed just by changing the wording a bit.
Thanks for doing this series!