Probability that someone has the disease

In summary, the conversation discusses the probability of someone having disease A based on the results of two tests with a 95% accuracy rate. The probability of having the disease if at least one test is positive is 0.1065 and if both tests are positive is 0.0115. The conversation also discusses the correct calculation for these probabilities and the importance of considering the independence of the two tests.
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

The percentage of people that have a disease A is $0,01$.
We apply twice a test for that disease, each of which give the correct answer with probability $0,95$.
What is the probability that someone has that disease if at least one test is positive and what is the probability if both tests are positive? I have done the following:

We have that $$P(\text{at least one positive})=1-P(\text{no positive})=1-P(NN)$$ where N: "negative". Let D be the event "has the disease".

Then we have that $$P(NN)=P(NN\cap D)+P(NN\cap D^C)=P(NN\mid D)\cdot P(D)+P(NN\mid D^C)\cdot P(D^C)$$ We have that $P(D)=0,01$ and $P(D^C)=0,99$.

Does it hold that $P(NN\mid D)=P(N\mid D)^2$ and $P(NN\mid D^C)=P(N\mid D^C)^2$ ?

If yes, does it hold that $P(N\mid D)=0,05$ and $P(N\mid D^C)=0,95$ ?

That would mean that we get $$P(NN)=0,05^2\cdot 001+0,95^2\cdot 0,99$$ Is this correct? (Wondering)
 
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  • #2
mathmari said:
Does it hold that $P(NN\mid D)=P(N\mid D)^2$ and $P(NN\mid D^C)=P(N\mid D^C)^2$ ?
If the two tests are independent, yes.

If yes, does it hold that $P(N\mid D)=0,05$ and $P(N\mid D^C)=0,95$ ?
Yes.

That would mean that we get $$P(NN)=0,05^2\cdot 001+0,95^2\cdot 0,99$$ Is this correct? (Wondering)

Except for the typo where you meant "0.01" and not "0.001", yes.

You know you have not yet answered either question, right?
 
  • #3
tkhunny said:
Except for the typo where you meant "0.01" and not "0.001", yes.

Ah ok!

So, we have that $$P(NN)=0.05^2\cdot 0.01+0.95^2\cdot 0.99=0.8935$$
tkhunny said:
You know you have not yet answered either question, right?

For the first question we have the following:
$$P(\text{at least one positive})=1-P(\text{no positive})=1-P(NN)=1-0.8935=0.1065$$
Is this correct?

At the suggested solution the result is $0, 093661972$. Can that be? (Wondering) For the second question we have the following:
\begin{align*}P(PP)&=P(PP\cap D)+P(PP\cap D^C) \\ & =P(PP\mid D)\cdot P(D)+P(PP\mid D^C)\cdot P(D^C)\\ & =P(P\mid D)^2\cdot P(D)+P(P\mid D^C)^2\cdot P(D^C)\\ & =0.95^2\cdot 0.01+0.05^2\cdot 0.99 \\ & =0.0115\end{align*}

Again the suggested result is different. It is $0, 784782609$. (Wondering)
 
  • #4
Isn't the question for P(D|at least one positive) respectively P(D|both positive)? (Wondering)
 
  • #5
I like Serena said:
Isn't the question for P(D|at least one positive) respectively P(D|both positive)? (Wondering)

Ah yes!

So, we have the following: $$P(D|\text{at least one positive})=\frac{P(D\cap \text{at least one positive})}{P(\text{at least one positive})}=\frac{P( \text{at least one positive}\mid D)\cdot P(D)}{0.1065}=\frac{P( \text{at least one positive}\mid D)\cdot 0.01}{0.1065}$$ Is everything correct so far? To what is $P( \text{at least one positive}\mid D)$ equal? (Wondering)
 
  • #6
Yep! And P(at least one positive|D)=1-P(NN|D).
 
  • #7
I like Serena said:
Yep! And P(at least one positive|D)=1-P(NN|D).

I see! Thank you! (Smile)
 

FAQ: Probability that someone has the disease

1. What is the definition of "probability"?

Probability is a measure of the likelihood that a specific event will occur. In the context of someone having a disease, probability would represent the chance that an individual has the disease.

2. How is the probability that someone has a disease calculated?

The probability that someone has a disease is calculated by dividing the number of individuals who have the disease by the total number of individuals in the population. This number is often expressed as a percentage or decimal value.

3. What factors influence the probability that someone has a disease?

The probability that someone has a disease can be influenced by a variety of factors such as age, gender, family history, lifestyle choices, and environmental factors. These factors can increase or decrease an individual's likelihood of having a disease.

4. Can the probability of someone having a disease change over time?

Yes, the probability of someone having a disease can change over time. This can occur due to changes in the individual's health, lifestyle, or exposure to risk factors. Additionally, advancements in medical technology and treatments can also impact the probability of someone having a disease.

5. Is probability the same as certainty?

No, probability and certainty are not the same. Probability represents the likelihood of an event occurring, while certainty refers to something being definite or guaranteed. In the context of someone having a disease, probability indicates the chance that an individual has the disease, while certainty would indicate a confirmed diagnosis.

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