The teacher led the student to a blind vendor who sold two types of incense, identical in appearance.
“This one he calls Forest,” the teacher said, holding up a speckled brown bundle that smelled of sandalwood and pine.
“And this one he calls Tea Garden,” the teacher said, holding up another.
The student sniffed the Tea Garden. “It has no odor,” she said.
“Ah, a shame,” the teacher said. “Only some people can smell Tea Garden.”
The student shrugged. “Well, let’s not buy any of that one.”
“He sells them in bags of two,” the teacher continued. “But he does not pay attention to which incense goes in which bag.”
The blind vendor had a massive crate of incense, with the two types mixed together. His hands dove in and out, grabbing another stick with each motion, and throwing it into a plastic bag. When a bag contained two sticks, he set it aside for sale.
“I’ll just smell the bags before we buy them,” the student said. “And we’ll only buy ones where I detect Forest.”
“Ah,” the teacher said. “But won’t we end up paying for lots of Tea Garden, too?”
“The bags we’ll pick already have one Forest,” the student said. “So there’s a 50% chance they’ll contain another Forest, and a 50% chance they’ll contain a Tea Garden. That seems worth the risk.”
“Then let’s buy 120 bags,” the teacher said. “Start choosing.”
When they got home, the teacher put her to work immediately. “Smell each stick of incense, one by one,” she said. “Then make two piles of bags. The first is for those with two Forest. The second is for those with only one Forest.” The teacher smiled. “And we shall see which pile is larger.”
“They’ll be the same size,” the student said. “Like I said—there’s a 1 in 2 chance the second stick is Forest, and a 1 in 2 chance that it’s Tea Garden.”
“But how do you know which is the second stick?” The teacher giggled and walked off.
By the time she finished sorting, the student had grown convinced something was wrong. The first pile—with two-Forest bags—held only 41 bags. The other pile—the one-Forest, one-Tea Garden bags—stood almost twice as high, with 79 bags.
“The vendor cheated us,” the student seethed. “He deliberately gave me extra Tea Garden,” the student said. “Only half of our bags should have Tea Garden. But instead, 2 out of 3 do.”
They returned the next day to the vendor’s stall. “Don’t just watch him,” the teacher said. “Help fill the bags yourself. You must see the incense as the vendor does, not as the customer does.”
So the student began to fill bags, smelling each stick of incense as she grabbed it from the crate, and noting the order.
After half an hour, she exclaimed suddenly, “I get it! There are four types of bags.”
“Yes. There’s Forest plus Forest. And Tea Garden plus Tea Garden. And then, there are two more possibilities. There’s Tea Garden plus Forest, and there’s Forest plus Tea Garden.”
“Those last two,” the teacher said, “aren’t they the same?”
“The bags look the same when you’re done,” the student said, “but they’re not created the same. The process for each one is different.
“When we came yesterday,” the student continued, “we eliminated one of the four possible bags—the ones with only Tea Garden. That left three other types of bags. So of the ones we brought home, the double-Forest should only be 1 in 3. I thought I’d only picked bags where the first stick was Forest, but that was wrong. In some of the bags I picked, Forest was the second stick.”
When they arrived home, the student lit a stick of incense and sat with her eyes closed. “This Forest smells different than it did yesterday,” she said.
The teacher smiled. “I’m not surprised,” she said. “That’s Tea Garden.”
This is a classic—and very tricky—problem. Usually, it’s presented the following way: “If a two-child family has at least one daughter, what is the probability that both their children are daughters?” Lots of people make the same mistake as the student in the story. The problem is an interesting introduction to the idea of “sample space,” because it requires careful thinking but almost no computation.
Sample space, by the way, is nothing but a fancy term for “list of possibilities.” The trick is that you’ve got to list possibilities that are equally likely. The student’s original sample space—F + F, TG + TG, and “one of each”—was flawed, because the last item was twice as likely as either of the other two.
Technically, the incense scenario is slightly different from the daughter scenario. If you have a daughter, the probability your next child is also a daughter is precisely 50%. But if our crate of incense starts out with a 50-50 mix, then once we pick out a stick of Forest, there’s less Forest than Tea Garden remaining. So our probability is just below 50%. (Luckily, if there are thousands of sticks of incense, as in the story, then this change is barely noticeable.)
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 happenin’ dude.