On Saturday morning, I reached a radiology in a neighbouring suburb for a dental x-ray at 9:30, hoping to wrap up the visit in 15 minutes. I ended up waiting for an hour - twice punctuated by my inquiries about expected waiting time - before an amiable, bespectacled radiologist materialised and led me to the x-ray room.
When asked why I had to wait so long when this type of x-ray should be a fairly short affair, he only murmured, "I don't know. It was a misunderstanding."
Back at the main waiting room, I handed the slip of paper that the radiologist had asked me to hand in to the reception, repeating to one of the receptionists what the radiologist had told me, that I had waited an hour as a result of a "misunderstanding".
The receptionist conferred with a colleague to her right and said something like, "You'd to wait as long as you did because the x-ray room has another machine that was being used by another patient for a procedure that takes time."
"You're unlucky," she added as she apologized and wished me a great day.
***
To begin with, what us lay folks call being "unlucky", experts from quantitative disciplines such as economics, social sciences and statistics may call being victims of "outliers", which are nothing but rare, out-of-the-ordinary events.
Examples of outliers include winning a lottery, the volcanic eruption of Mt. Vesuvius in 79 AD that buried Pompeii, air crashes in developed nations, Australia's national rugby team Wallabies posting a win against New Zealand's All Blacks ... and, apparently, waiting for an hour to get one's "missing/crowded" teeth x-rayed.
Since my dentist indicated that I will have to make a number of visits to the radiology, I am mainly concerned with two questions.
First, what is the probability of my being "unlucky" in the same radiology's waiting room in my next visit?
For the sake of argument, let's assume that, on average, 1 in 100 dental x-ray patients on a Saturday morning gets "unlucky" in that particular radiology, ending up waiting an hour. This means the probability of getting unlucky is 1 percent or 0.01.
From this, assuming that two visits to the radiology for dental x-rays are independent, i.e. the first visit does not influence the waiting time of the second visit, the probability that I will again be "unlucky" on the second visit is still 0.01.
A slightly different question is this. What is the probability that a patient like myself will be "unlucky" in two consecutive visits? Intuition tells us that it has to be lower than 1 in 100. In fact, it is 0.01 x 0.01 = 0.0001 or 1 in 10,000.
On the other hand, I may reason that since I was already unlucky in my last visit to the radiology, the probability of getting unlucky in the next visit is lower than 1 in 100. If I reason like this, assuming that the previous visit has no effect on the waiting time of my next visit, I have just fallen victim to the gambler's fallacy.
The next question I am interested is this. If I am not unlucky in my next visit to the radiology, what should I expect the waiting time to be? To put it another way, what is the average waiting time for a dental x-ray on a Saturday morning?
Assuming that waiting times are normally distributed and a waiting time of 1 hour lies in the upper 1 percent of the distribution, i.e. 99 percent of waiting times are less than 1 hour, the average waiting time is around 35 minutes, with the standard deviation of around 11 minutes.
This means that my initial hope of wrapping up the visit in 15 minutes was a forlorn hope. From the properties of the normal distribution, it can be estimated that there is less than 3 percent likelihood of a waiting time to be 15 minutes or less.
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