I’m currently teaching a unit on probability. One of the homework problems that I wrote was roughly follows. Consider the experiment of drawing four cards without replacement from a thoroughly shuffled deck of cards. What is the probability that the first card is a heart, the second is a diamond, the third is a heart, and the fourth is a spade?
A student, Michael, came into my office for help on the homework problems. While he was asking about various problems, I was shuffling a deck of cards. When we got to the problem given above, I first asked him for his intuition about the problem –was the probability likely or unlikely. He recognized that it was extremely unlikely. Then I used the deck of cards to illustrate the problem. So, I draw a card from the deck. I asked, “What’s the probability that it’s a heart?” “13/52” he answered. I drew a card from the top of the deck. It was a heart. “How about that, it was actually a heart,” I said. “What’s the probability that the next card is a diamond?” “13/51”. I drew the next card. It was a diamond. Wow. “Okay, what’s the probability that the next card is a heart?” “12/50”. I drew the next card. It was a heart! At this point we were pretty surprised. Michael made some comment about he really hoped the next card wouldn’t be a spade. I drew the next card . . . it was a spade! Weird.
The probability of this happening . . . about 0.4%.
Of course, this far from the first time that I had illustrated an experiment like this, so it’s not so surprising that eventually a coincidence would occur eventually. But still, it was pretty exciting.