'Deadly Disease' Challenge With Maths

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A-level Maths By: Tutor no longer registered
Subject: Maths » A-level Maths
Last updated: 04/06/2017
Tags: a level and gcse maths, a-level statistics, probability

‘You hear about a deadly disease which has no symptoms but is absolutely fatal. Only 1 in 10,000 people have this terrible disease but you decide that to be on the safe side you should see your Doctor and take the test anyway.

Your Doctor explains that the test is 99% accurate and that you will get your results in two weeks. In two weeks’ time the letter arrives and it says you have the disease. Should you be worried?’

I find that maths can come alive for students when real world examples are used and this is especially prominent when the solution is counter-intuitive.

This type of question can be used for a range of ages studying mathematics but with varying levels of scaffolding.

This question is best illustrated using a table and if we suppose 1,000,000 people are tested for this deadly disease, we can start completing the table by filling in how many of these people are sick (column 1). If 1 in 10,000 people have this disease, of the 1,000,000 tested, 100 have it. Now, since the test is 99% accurate, 99 of these people tested positive and 1 tested negative.

Now, if we consider the healthy people (column 2), 1,000,000 – 100 = 999,900 don’t have the disease. Of the healthy people, 99% would have tested negative (correctly). This equates to 999,900 x 0.99 = 989,901 healthy people with the correct negative result. However, this leaves 9999 healthy people who tested positive for the deadly disease.

 

Sick

Healthy

Totals

Tested Positive

99

9,999

10,098

Tested Negative

1

989,901

989,902

Totals

100

999,900

1,000,000


From this chart, the answer to our original question can easily be found. You have tested positive for the deadly disease so using conditional probability we are concerned with the chances of having the disease given the test result is positive. So, we only care about row 1.

The probability of having the disease given you tested positive is 99 out of 10,098 = 1%

The probability of not having the disease given you tested positive is 9,999 out of 10,098 = 99%

Therefore, you should not be worried at all!

It seems quite hard to believe but the rarer the disease, the lower the probability that a positive result means you have it!

The exact probabilities will vary depending on the accuracy of the test and the actual incidence of the disease, but you always must look at the conditional probability.

This is one reason why, for a disease like AIDS, patients are never told they test positive until the blood has been retested with a different test, to minimize the chance of a false positive

Doctors should be familiar with the probabilities involved in testing for rare diseases but if you ever get a positive result for a rare disease on the first test, you can rest assured that you don’t have to worry… probably.





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