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It’s never lupus. Lupus is a medical phenomenon where antibodies that are supposed to attack foreign cells to prevent infections instead see plasma proteins as foreign bodies, leading to a high risk of blood clotting. It is believed that 2% of the population suffer from this disease. The test is 98% accurate if a person actually has the disease. The test is 74% accurate if a person does not have the disease. There is a line from the Fox television show House that is often used after a patient tests positive for lupus: “It’s
never lupus.” Do you think there is truth to this statement? Use appropriate probabilities to support your answer.

Sagot :

L = person has Lupus

~L = person does not have Lupus

P(L) = 0.02 since 2% of the population has it.

P(~L) = 1 - 0.02 = 0.98, ie 98% of the population does not have it.

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If the person has Lupus, then the test will indicate as such 98% of the time, and do so correctly.

T = person tests positive for Lupus

~T = person tests negative for Lupus

P(T given L) = 0.98

The complement of this is

P(~T given L) = 1 - 0.98 = 0.02 = false negative rate

On the other hand, if a person does not have Lupus, then the test will return negative correctly 74% of the time.

P(~T given ~L) = 0.74

Its complement is

P(T given ~L) = 1 - 0.74 = 0.26 = false positive rate

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Dr House saying "It's never Lupus" can be translated to "If the test is positive, then P(L) = 0"

Phrased another way symbolically:

P(L given T) = 0

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We'll use Bayes Theorem to determine the value of P(L given T)

In general, Bayes Theorem is

P(A given B) = P(B given A)*P(A)/P(B)

So for this problem, we'll have

P(L given T) = P(T given L)*P(L)/P(T)

We almost have everything needed but we're missing P(T)

So we'll need to use the law of total probability to break that up like so

P(T) = P(T and L) + P(T and ~L)

P(T) = P(T given L)*P(L) + P(T given ~L)*P(~L)

P(T) = 0.98*0.02 + 0.26*0.98

P(T) = 0.2744

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To recap, we have:

  • P(T given L) = 0.98
  • P(L) = 0.02
  • P(T) = 0.2744

We finally have all the info needed to get the probability we're after.

P(L given T) = P(T given L)*P(L)/P(T)

P(L given T) = 0.98*0.02/0.2744

P(L given T) = 0.07142857142858 approximately

What does this tell us? It says that if a patient gets a positive test for lupus, then there's roughly a 7% chance the test is correct, and the person actually has lupus.

So it's likely Dr House is onto something. The reason why the small result is because a small percentage of the population has the disease.

Also, notice how the false positive rate P(T given ~L) = 0.26 = 26% is fairly high. This also provides more evidence to Dr House's claim.

To be entirely technically accurate, he shouldn't have used "never" because the result of P(L given T) is not zero. But I think a bit of wiggle room can be had for the interpretation of his words.

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