some truth behind the untruth

First, let’s say there are only two of you in a room. What is the chance that you have only one birthday? Let’s say yours is May 20. There are 364 ways that your friend Romi’s birthday is different from yours, and only 1 (May 20) in which it is the same. So the probability that you share a birthday is 1/365, or 0.0027. Now your friend Parth comes in the room. There are 364 ways that Romi’s birthday is different from yours and 363 ways that Parth’s birthday is different from yours. Thus the probability of all three having different birthdays is 364/365 x 363/365 = 0.9918. So the chance of at least two of you sharing a birthday is 1 – 0.9918 = 0.0082. Still very young, but better than the time it was just you and Romy.

Continue the same logic for 4 and 5 and 6 people… and yes, when you have 23 people in your room, the probability that at least two have the same birthday crosses 50%. So the next time you’re in that size group, give it a try. The thing is, the implications of coincidence and probability can often take you by surprise.

I’ve been thinking about it lately. Someone I know called Ludhianvi flew to the UK last week. He had to undergo an RT-PCR test before leaving – which of course needed to be negative. On arrival, UK government regulations required him to go into home quarantine for 10 days with RT-PCR tests on day 2 and 8. Then, both have to be negative to exit the quarantine after 10 days.

Indeed, Ludhianvi’s pre-departure test came back negative, as did his second day’s test. He is now counting down his last few days in quarantine, itching to get out. She’s with a family that’s been fully vaccinated, and she hasn’t met anyone else yet – another reason she’s itching to step out.

Imagine her surprise when she suddenly received an email from the UK’s National Health Service last Monday. It said: “[We have] You have been identified as a contact of someone who has recently tested positive for COVID-19. Now you must stay at home and self-isolate for 10 days from the date of your last contact with them.”

How did this happen? Apart from the family he is with, Ludhianvi has had no “contact” for over a week, who was now his fellow-passenger on the London flight. Clearly, one of them must have tested positive two days after landing, and that is the reason for the message from the NHS. Yet the passengers were all like him, with the airline only being allowed to fly after showing a negative test result. That is, they were all certified virus-free. How is it possible that one of these passengers tested positive two days after landing?

Two phrases to remember here: “false negatives” and “false positives”. That is, one or the other of this person’s two tests—before flight, and on the second day of quarantine—gave a false result. Either the pre-flight test was negative when this person actually had the virus – a “false negative” – ​​or the second day’s test was positive when they didn’t actually have the virus – a “false positive”.

What is the probability of such a mistake? Of course, it is less. These are generally reliable tests. But they are not foolproof. For example, “a systematic review reported false-negative rates of between 2% and 33%” and “preliminary estimates show [the false-positive rates] May be somewhere between 0.8% and 4.0%.” (False-positive COVID-19 Results: Hidden Problems and Costs, Elena Surkova et al., The Lancet, 29 September 2020, bit.ly/3EmjwPC).

Let’s take the lowest number of them: 0.8%. Meaning, the second day’s test gives a positive result for someone who is not infected, is 0.8%, or 1/125. That is, the probability of this person getting a true negative result is 124/125, or 0.992, or close to certain.

By comparison, the probability of two people having the same birthday is 1/365, or about 0.27%. Remember the calculation above that came up with 23 people? In roughly the same way, we can calculate how many test takers we need to have a better than 50% probability that at least one gets a false-positive result.

Is it 2 students? The probability that both the tests are negative is 0.992 x 0.992 = 0.984. Thus the probability of getting at least one false positive is 1 – 0.984 = 0.016. 3 people? The probability that all three tests are negative is 0.992 x 0.992 x 0.992 = 0.976. So the probability of at least one of those false positives is 1 – 0.976 = 0.024.

4 people? 0.032.

5 people? 0.04.

…

87 people? Ahh! 0.503.

That is, when 87 people who do not have the virus take a test that has a false-positive rate of 0.8%, the chances are about 50-50 that one of them will test positive anyway.

Now of course there were over 87 people on the flight to London. In fact, it is likely that there were over 200 passengers, all carrying negative pre-flight test reports. So if 200 UK visitors from the Ludhianvi flight tested positive on the second day, we could do the same calculation to estimate the probability of getting at least one false positive. That chance: 80%. which is close to certainty.

In fact, there is more to all of this. For example, what are the implications of a pre-flight false-negative report? This would mean that someone in flight was indeed infected. Again, given the false-negative probabilities and the numbers on the plane, this is very likely to happen, and would make it certain that Ludhianvi was in “contact” with someone who tested positive.

Furthermore, a thorough study of the numbers suggests that “low prevalence of the virus” – which is the case in the UK – in a population produces “a significant proportion of false-positive results”. This “adverse effect”[s] Positive predictive value of the test. (Quotes from the same paper cited above). Again, its implications are worth understanding.

Still, to no one should be surprised that Ludhianvi got his message from the NHS. In fact, the numbers suggest that almost everyone who flies into the UK and goes into quarantine will, more than likely, receive such a message. Because on any flight carrying 200 or more passengers – and modern long-range jets can pack in 250 or more – it is almost certain that at least one will test positive after landing.

To liven up these Covid days, I think the message the NHS should send is: “We have identified you as a contact of two people who share the same birthday.”

Dilip D’Souza, once a computer scientist, now lives in Mumbai and writes for his dinner. His Twitter handle is @DeathEndsFun