The following question appeared in the “Ask Marilyn” column in the August 16, 2015 issue of Parade magazine. The writer seems stuck between two probabilities.
(Click here for a cleaned-up online version if you don’t like the newspaper look.)
I just pitched this question to my statistics class (we start the year with a probability unit). I thought some of you might like it for your classes, too.
I asked them to do two things. 1) Answer the writer’s question, AND 2) Use precise probability terminology to identify the source of the writer’s conundrum. Can you answer both before reading further?
Very briefly, the writer is correct in both situations. If each of the four people draws a random straw, there is absolutely a 1 in 4 chance of each drawing the straw. Think about shuffling the straws and “dealing” one to each person much like shuffling a deck of cards and dealing out all of the cards. Any given straw or card is equally likely to land in any player’s hand.
Now let the first person look at his or her straw. It is either short or not. The author is then correct at claiming the probability of others holding the straw is now 0 (if the first person found the short straw) or 1/3 (if the first person did not). And this is precisely the source of the writer’s conundrum. She’s actually asking two different questions but thinks she’s asking only one.
The 1/4 result is from a pure, simple probability scenario. There are four possible equally-likely locations for the short straw.
The 0 and 1/3 results happen only after the first (or any other) person looks at his or her straw. At that point, the problem shifts from simple probability to conditional probability. After observing a straw, the question shifts to determining the probability that one of the remaining people has the short straw GIVEN that you know the result of one person’s draw.
So, the writer was correct in all of her claims; she just didn’t realize she was asking two fundamentally different questions. That’s a pretty excusable lapse, in my opinion. Slips into conditional probability are often missed.
Perhaps the most famous of these misses is the solution to the Monty Hall scenario that vos Savant famously posited years ago. What I particularly love about this is the number of very-well-educated mathematicians who missed the conditional and wrote flaming retorts to vos Savant brandishing their PhDs and ultimately found themselves publicly supporting errant conclusions. You can read the original question, errant responses, and vos Savant’s very clear explanation here.
Probability is subtle and catches all of us at some point. Even so, the careful thinking required to dissect and answer subtle probability questions is arguably one of the best exercises of logical reasoning around.
As a completely different connection, I think this is very much like Heisenberg’s Uncertainty Principle. Until the first straw is observed, the short straw really could (does?) exist in all hands simultaneously. Observing the system (looking at one person’s straw) permanently changes the state of the system, bifurcating forever the system into one of two potential future states: the short straw is found in the first hand or is it not.
CORRECTION (3 hours after posting):
I knew I was likely to overstate or misname something in my final connection. Thanks to Mike Lawler (@mikeandallie) for a quick correction via Twitter. I should have called this quantum superposition and not the uncertainty principle. Thanks so much, Mike.