Poser #1: Yesterday a colleague from another school asked me how to test whether hummingbirds would visit two related species of flowers nonrandomly, that is, whether the flowers were reproductively isolated because the hummingbird (which pollinates as it sips nectar) prefers one over the other. He proposed an experiment in which he would put two individual plants of each species in a four-plant array, and then watch which ones the hummingbirds visited. His initial supposition was that if there were no reproductive isolation (that is, the species are equivalently attractive to the bird), any two bird visits would result in the flowers being of different species 50% of the time. As he said,
“All else being equal, you’d expect a bird that’s falling from the sky (telling itself, ‘I am GOING to pick two random flowers’) the thing should visit two A’s 25% of the time, two B’s 25% of the time, and 50% of the time it should visit one of each species.
Again, it SHOULD pick two DIFFERENT species 50% of the time if it somehow chose them simultaneously.”
But then he realized that that wasn’t right. He asked me what the answer was, and, after about 5 minutes of thought, it came to me, and it’s obvious if you think about it.
This is equivalent to putting two black balls and two white balls in an urn, and then picking two balls. What are the chances that you’d pick two balls of different color?
It’s not 50%. And the true answer doesn’t matter whether you draw the balls successively, or grab two balls at once.
What’s the answer? (It’s the same as if the department has four new graduate students: two males and two females, and asks you to put two of them in a vacant, two-person office. What are the chances they’d be of opposite sex if you choose randomly?
Explain your answers below.
And this real-world biological problem brought to mind a very famous hypothetical problem:
Poser #2. This is the most counterintuitive probability poser I know: the famous “Monty Hall” problem, about which my pal Jason Rosenhouse wrote a whole book. It’s based on the old television game show, “Let’s Make a Deal,”hosted by Monty Hall, which gave contestants a choice like the one described below.
Here’s the deal: You are shown three doors. Behind one of them is a fabulous prize, like a car or a vacation. Behind the other two are trivial prizes, say pillows. You choose a door. The host, who knows what’s behind every door, then opens one of the doors you didn’t choose, revealing a pillow. He then asks you, “Do you want to switch doors now?” That is, he’s saying you can stick with the door you originally chose, or switch to the other, unopened one. Whatever you decide to do, you get what’s behind the door you stick with at the end.
The question: should you switch doors? The intuitive answer is “no, it doesn’t matter: the chance I’ll choose the one with the prize is 50% whether I switch or not.”
That’s the wrong answer. It does matter.
I’m sure some of you know the answer, but wait a while before you explain it in the comments. For those who can’t wait, the explanation is here.