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Home >Opinion >Columns >Some truth behind what’s false

Some truth behind what’s false

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The implications of chance and probability can often surprise you

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There’s a simple but startling trick you can indulge in any time you’re in a gathering of about 25 people (or more). It goes like this: pull out a suitably high-denomination currency note and slap it on the table. Announce that as your stake in this bet: that there will be at least two people in the group who have the same birthday—meaning not the year, but just the day, like 17 March, or 2 October. I assure you, this is a good bet. Or at least, you have a better-than-even chance of winning it. The reason? When your gathering reaches 23 people, the chance of finding such a pair rises past 50%. If that seems counter-intuitive, a slice of fairly simple arithmetic will confirm it. Consider:

There’s a simple but startling trick you can indulge in any time you’re in a gathering of about 25 people (or more). It goes like this: pull out a suitably high-denomination currency note and slap it on the table. Announce that as your stake in this bet: that there will be at least two people in the group who have the same birthday—meaning not the year, but just the day, like 17 March, or 2 October. I assure you, this is a good bet. Or at least, you have a better-than-even chance of winning it. The reason? When your gathering reaches 23 people, the chance of finding such a pair rises past 50%. If that seems counter-intuitive, a slice of fairly simple arithmetic will confirm it. Consider:

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

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

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

I’ve been thinking of that in recent days. Someone I know, call him Ludhianvi, took a flight to the UK last week. He had to have a RT-PCR test before leaving—which needed to be negative, of course. On arrival there, UK government rules required him to go into home quarantine for 10 days, with RT-PCR tests on the second and eighth days. Again, both need to be negative for him to exit quarantine after the 10 days are done.

Indeed, Ludhianvi’s pre-departure test came back negative, as did his second-day test. He’s now counting down his last few days in quarantine, itching to get out. He’s with a family that’s fully vaccinated, and he has not been able to meet anyone else so far—another reason he’s itching to get out.

Imagine his surprise, then, when out of the blue last Monday, he got email from the UK’s National Health Service. It said: “[We have] identified you as a contact of someone who has recently tested positive for covid-19. You must now 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’s with, the only people Ludhianvi has had any “contact" with for over a week now were his fellow-passengers on the flight to London. Clearly, one of them must have tested positive two days after landing, and that’s the reason for the message from the NHS. Yet the passengers were all like him, allowed to take the flight only after showing the airline a negative test result. That is, they were all certified virus-free. How is it possible that one of the passengers tested positive two days after landing?

Two phrases to remember here: “false negative" and “false positive". That is, one or the other of this person’s two tests—pre-flight, and on the second day of quarantine—produced an erroneous result. Either the pre-flight test was negative when this person actually had the virus—a “false negative"—or the second-day test was positive when they actually did not have the virus—a “false positive".

What’s the chance of such a mistake? Admittedly, it’s low. These are generally reliable tests. But they are not foolproof. For example, “one systematic review reported false-negative rates of between 2% and 33%" and “preliminary estimates show [the false-positive rates] could be somewhere between 0.8% and 4.0%." (False-positive COVID-19 results: hidden problems and costs, Elena Surkova and others, The Lancet, 29 September 2020, bit.ly/3EmjwPC).

Let’s take the lowest of those numbers: 0.8%. Meaning, the chance that the second-day test returns a positive result for someone who isn’t infected is 0.8%, or 1/125. That is, the chance this person gets a correct negative result is 124/125, or 0.992, or close to certain.

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

Is it 2 test-takers? The chance that both test negative is 0.992 x 0.992 = 0.984. Thus the chance that at least one gets a false positive is 1 - 0.984 = 0.016. 3 people? The chance that all three test negative is 0.992 x 0.992 x 0.992 = 0.976. So the chance of at least one false positive among them is 1 - 0.976 = 0.024.

4 people? 0.032.

5 people? 0.04.

...

87 people? Aha! 0.503.

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

Now certainly the flight to London had more than 87 people. In fact, it’s likely there were over 200 passengers, all carrying negative pre-flight test reports. So if 200 visitors to the UK from Ludhianvi’s flight took the second-day test, we can do the same calculation to estimate the chance that at least one gets a false positive. That chance: 80%. That close to certainty.

There’s more to all this, actually. What are the implications of a pre-flight false-negative report, for example? That would mean that someone on the flight actually was infected. Again, given the false-negative probabilities and the numbers on the plane, that’s very likely to have happened, and that would make it even closer to certain that Ludhianvi was in “contact" with someone who tested positive.

Besides, a deeper perusal of the numbers shows that a “low prevalence of the virus"—which is the case in the UK—in a population produces “a significant proportion of false-positive results". This “adversely affect[s] the positive predictive value of the test." (Quotes from the same paper cited above). Again, the implications there are worth understanding.

Still, it should surprise nobody that Ludhianvi got his message from the NHS. In fact, the numbers suggest that nearly everyone who flies into the UK and goes into quarantine will, more than likely, get such a message. Because on any flight carrying 200 or more passengers—and modern long-distance jets can pack in 250 or many more—it’s nearly certain that at least one will test positive after landing.

Just to liven up these covid days, I think the NHS should send out this message: “We have identified you as a contact of two people who share the same birthday."

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

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