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MMM #22 Winner

The winner for this edition of MMM is Henno Brandsma. Congratulations Henno on solving the problem and calming my nerves that I have less than 1% chance of having the virus. In fact almost everyone who submitted (23 this time) got it correct! Maybe we need more difficult problems like the Martian Problem or our most popular problem Mac Madness. Give us some feedback on ideas and promotions. And thanks again for everyone who were creative in their prescriptions and calming my nerves. We enjoyed reading them!

The Answer by Henno Brandsma

Henno is an editor for Ask a Topologist. Check it out if you are interested in topology, geometry, analysis, and algebra.

Let’s define some events:
Let G = a person has the Grinch virus, and T = the test is positive.
Given P(G) = 0.0001 (1 in 10000 has the virus), P(T | G) = 0.99 = P( not T | not G).
So P(not T | G) = 1 - 0.99 = 0.01, P (T | not G) = 1 - P(not | not G) = 0.01.
Also P(not G) = 0.9999

It follows that P(T) = P(T | G)*P(G) + P(T | not G)*P(not G) =

0.99 * 0.0001 + 0.01*0.9999 = 0.010098

Asked for: P( G | T).

P(G | T) = P ( G /\ T) / P(T) = P( T | G)*P(G) / P(T) =

0.99 * 0.0001 / 0.010098 = 0.0098..

so less than 1% chance that he has the virus.

Denise at Let’s Play Math! blog

Alternatively, an intuitive method is shown by Denise from Let’s Play Math!

Since the percentage would be the same for any numbers, we will assume an easy-to-work-with number of people are tested for the Grinch Bug:

People tested: 1,000,000.

People who really have the bug:
1,000,000 / 10,000
= 100.
Number who have the bug, who test positive (true test result):
= (99/100) * 100
= 99.

People who don’t have the bug:
1,000,000 - 100
= 999,900.
Number who don’t have the bug, who test positive (false test result):
= (1/100) * 999,900
= 9,999.

Total number of positive test results:
= 9,999 + 99
= 10,098.

Probability that the positive test result means you really are sick:
= 99 / 10,098
= 0.0098…
~ 0.98 %.

There is less than a 1% chance that you have the disease, despite the positive test result. The doctor was correct — still no need to worry.

Follow this math teacher’s prescription: Fix a big mug of hot chocolate (use a candy cane for stirring), and settle down with a book of math puzzles. You’re sure to feel better after that!