Here’s an interesting math puzzle involving probability…
1 out of every 10,000 citizens of a country has a deadly virus. A blood test can be used to determine whether or not somebody has the virus. The test is accurate 98% of the time. There are 1 million people in the country, and each person is tested. A man who tested positive for the virus talks to his doctor, who informs the man that there is a 98% chance that he does, in fact, have the virus. The doctor is not correct in making this statement… what is the actual probability that the man has the deadly virus?
To see the answer, scroll down to the bottom. If you would like help in understanding the logic behind the solution, keep reading.
First of all, we have to understand what probability is.
We know that 1 out of every 10,000 people actually have the virus.
Therefore, out of 1 million people, 100 people have the virus. (1,000,000 ÷ 10,000 = 100)
The probability that a citizen of the country has the virus is 100 ÷ 1,000,000 = 0.0001 = 0.01%.
This probability is independent of the blood test.
Once the blood test is administered to every citizen in the country, the question becomes a bit more difficult. We need to determine the probability that a person who tested positive for the virus does, in fact, have the virus.
So, how many people tested positive for the virus?
98% of people who have the virus tested positive. 98% of 100 is 98.
If 100 people have the virus, then 999,900 people do not have the virus. (1,000,000 – 100 = 999,900)
98% of people who don’t have the virus tested negative. This means that 2% of people who don’t have the virus tested positive. 2% of 999,900 is 19,998.
The total number of people who tested positive is 98 + 19,998 = 20,096.
Now we can calculate the probability that the man who tested positive does, in fact, have the virus.
20,096 people tested positive for the virus. Of these, 98 people actually have the virus.
Therefore, the probability that the man has the virus is 98 ÷ 20,096 = 0.0049 = 0.49%.
That’s less than half of one percent!
When considering this fact, it is tempting to think that the test is practically worthless – but this is not the case.
Before the test was administered, all we knew was that 0.01% (one 100th of a percent) of the population had the virus. Now a group has been isolated in which about half of a percent has the virus.
If this group is re-tested, the probability that a person testing positive does, in fact, have the virus will be about 24%.
If the group that tests positive a 2nd time is re-tested, the probability that a person testing positive does, in fact, have the virus will be about 92%.
It’s amazing to see how quickly accuracy is gained through the process of re-testing.
And now you know a little bit of the mathematics that goes into being a good doctor.
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