# Real life probability riddle

#### eeisen

##### New Member
First time poster. I have something of a probability riddle that I'm not sure how probability experts would think about it. The story is as follows:

My spouse and I received a second dose of the pfizer vaccine on Jan 14 and traveled to the USA on Jan 17. Our four children did not receive the vaccine, and no one in the family has any indication of having covid previously. Due to covid restrictions, we were unable to return home (outside of USA) until Feb 23. On Feb 15, my wife and I, with our two older children took antibodies tests; my wife and I had confirmed antibodies from the vaccine, my children did not have antibodies. In order to fly, we all had to take a PCR test within 3 days of flying, which we did at a reputable lab and received negative results. Upon arrival to our home country, we took another PCR test in the airport, which yielded all of the children negative, but my wife and I were positive. My kids retested 2 days later and were all negative again, and my wife and I retested this morning and are awaiting results.
Let's give the following assumptions: they are testing 100,000 people per day, with 4% positive rate. Reports put false positive rates at somewhere between .8% and 3.5%. False negative rates are a little more challenging because of the way the infection develops. A recent study put the overall discordant rate at around 0.05%. We do not know the discordant rates for the antibodies test, but the results were consistent with our expectations based on our vaccine and no known exposure or infection over the course of the the past year?
My question for the group is what is the probability of both my wife and my own positive test after the flight being incorrect, and that whatever results we receive from today's test are correct?
And, to make it fun, would you bet that our test results today would be positive or negative?

#### fed2

##### Active Member
i like ur chances, pcr results could drop to negative in days, especially since you don't feel sick. could be on 'on the upswing' though, in which case you have my sympathy. im not sure the rates you give are very useful to your specific question about retesting, since these are really more about comparing indpendantly selected people more than retesting same peoples. you might find 'positive predicitve value' a useful google though.

#### eeisen

##### New Member
Thank you for the reply! The way I was thinking about it was assuming they were testing 100k, with 4% assumed to be positive in the population of those being tested, that means that there will be 96k negative and 4k positive. Assuming 1% false positive, and 20% false negative, that means there will be 960 false positive, and 3200 true positives, for about 23% of positives being false. With the negatives, there would be 95,040 true negatives, and 800 false negatives, for a 0.8% of negatives being false. Which means that for both my wife and I to have false negatives, the likelihood would be .007%, which for both to be false positives, it would be 5.3% likelihood, which makes it 764 times more likely to be false positives, given the assumptions.

I have no idea if that is a statistically sound way to approach this problem, but that's what I had been thinking about.

As it turns out, we received negative tests following, and the lab wrote back that they found the gene for covid positive at cycle 37, which in most labs would be considered "inconclusive" or not even run to that many cycles.

Stay safe everyone!

#### fed2

##### Active Member
yeah i think u got the logic right. the key thing is that the false positives depend both on the test characteristics and the population being tested. the disease is actually pretty rare, so even good tests have some false positives.