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.
I once gave a talk at a gathering of high school students competing in a math olympics contest, and the main thesis I wanted to prove to them was: It is very likely that rare events occur. We discussed the odds of a particular pattern of outcome when flipping a coin six times. Any particular pattern, we all agreed, was remote, occurring with probability 1/64. And then on the stage I actually flipped the coin six times. But it was manifest that we would get *some* pattern, so I had proved my point, that it was very likely that a rare event occurs.
We went on to look at the paradoxical situations that arise when someone gets a positive test result for a rare disease. Should they be worried? If the test is 99% accurate, but the disease occurs in say, 1 in million, then a positive result is not so worrisome: in million people, there will be about 1 true positive result, and about 10,000 false positives, since 1% of a million is 10,000. So the odds that you’ve actually got the disease, given that you tested positive, is 1 in 10,000.
But your situation is completely different! It would be as though we had calculated the odds of getting HHHTTT, and then when I actually flipped the coin on stage, i actually got the same pattern HHHTTT. Totally weird! And very unlikely. But you know, if it wasn’t that, it would have been some other totally unlikely thing, like getting all green lights, or all red lights, or getting the serial number 123456 on your receipt at Starbucks.