Solved on Nov 17, 2023

A virus infects 1 in 400 people. A test is positive 80% if infected and 10% if not infected. Find the probability a person has the virus given a positive test, and the probability a person doesn't have the virus given a negative test, rounded to the nearest tenth of a percent.
$P(A \mid B) = 95.2\%$ $P(A' \mid B') = 99.8\%$

STEP 1

Assumptions1. The probability of a person being infected by the virus, denoted as $(A)$ , is $\frac{1}{400}$ . . The probability of a person testing positive given they are infected, denoted as $(B|A)$ , is $80\%$ or $0.8$ .
3. The probability of a person testing positive given they are not infected (false positive), denoted as $(B|A')$ , is $10\%$ or $0.1$ .
4. The probability of a person not being infected by the virus, denoted as $(A')$ , is $1 -(A)$ .
5. We are looking for $(A|B)$ , the probability of a person being infected given they have tested positive, and $(A'|B')$ , the probability of a person not being infected given they have tested negative.

STEP 2

First, we need to calculate $(A')$ , the probability of a person not being infected by the virus. We can do this by subtracting $(A)$ from $1$ .
$(A') =1 -(A)$

STEP 3

Now, plug in the given value for $(A)$ to calculate $(A')$ .
$(A') =1 - \frac{1}{400}$

STEP 4

Calculate $(A')$ .
$(A') =1 - \frac{1}{400} = \frac{399}{400}$

STEP 5

Next, we need to calculate $(B)$ , the total probability of a person testing positive. We can do this using the law of total probability, which states that $(B) =(B|A)(A) +(B|A')(A')$ .

STEP 6

Now, plug in the given values for $(B|A)$ , $(A)$ , $(B|A')$ , and $(A')$ to calculate $(B)$ .
$(B) =0.8 \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400}$

STEP 7

Calculate $(B)$ .
$(B) =0. \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400} = \frac{40}{400} =0.1$

STEP 8

Now, we can calculate $(A|B)$ , the probability of a person being infected given they have tested positive. We can do this using Bayes' theorem, which states that $(A|B) = \frac{(B|A)(A)}{(B)}$ .

STEP 9

Now, plug in the given values for $(B|A)$ , $(A)$ , and $(B)$ to calculate $(A|B)$ .
$(A|B) = \frac{.8 \cdot \frac{}{400}}{.}$

STEP 10

Calculate $(A|B)$ .
$(A|B) = \frac{0.8 \cdot \frac{}{400}}{0.} =0.002 =0.2\%$

STEP 11

Next, we need to calculate $(B')$ , the total probability of a person testing negative. We can do this using the law of total probability, which states that $(B') =(B'|A)(A) +(B'|A')(A')$ .

STEP 12

We know that $(B'|A) = -(B|A)$ and $(B'|A') = -(B|A')$ . So, plug in the given values for $(B|A)$ and $(B|A')$ to calculate $(B'|A)$ and $(B'|A')$ .
$(B'|A) = -0.8 =0.2$ $(B'|A') = -0. =0.9$

STEP 13

Now, plug in the calculated values for $(B'|A)$ , $(A)$ , $(B'|A')$ , and $(A')$ to calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400}$

STEP 14

Calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400} =0.9$

STEP 15

Finally, we can calculate $(A'|B')$ , the probability of a person not being infected given they have tested negative. We can do this using Bayes' theorem, which states that $(A'|B') = \frac{(B'|A')(A')}{(B')}$ .

STEP 16

Now, plug in the calculated values for $(B'|A')$ , $(A')$ , and $(B')$ to calculate $(A'|B')$ .
$(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9}$

STEP 17

Calculate $(A'|B')$ .
$(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9} =0.9975 =99.75\%$ So, the probability that a person has the virus given that they have tested positive is $0.2\%$ and the probability that a person does not have the virus given that they test negative is $99.75\%$ .

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STEP 1

STEP 2

First, we need to calculate $(A')$ , the probability of a person not being infected by the virus. We can do this by subtracting $(A)$ from $1$ .
$(A') =1 -(A)$

STEP 3

Now, plug in the given value for $(A)$ to calculate $(A')$ .
$(A') =1 - \frac{1}{400}$

STEP 4

Calculate $(A')$ .
$(A') =1 - \frac{1}{400} = \frac{399}{400}$

STEP 5

Next, we need to calculate $(B)$ , the total probability of a person testing positive. We can do this using the law of total probability, which states that $(B) =(B|A)(A) +(B|A')(A')$ .

STEP 6

Now, plug in the given values for $(B|A)$ , $(A)$ , $(B|A')$ , and $(A')$ to calculate $(B)$ .
$(B) =0.8 \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400}$

STEP 7

Calculate $(B)$ .
$(B) =0. \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400} = \frac{40}{400} =0.1$

STEP 8

Now, we can calculate $(A|B)$ , the probability of a person being infected given they have tested positive. We can do this using Bayes' theorem, which states that $(A|B) = \frac{(B|A)(A)}{(B)}$ .

STEP 9

Now, plug in the given values for $(B|A)$ , $(A)$ , and $(B)$ to calculate $(A|B)$ .
$(A|B) = \frac{.8 \cdot \frac{}{400}}{.}$

STEP 10

Calculate $(A|B)$ .
$(A|B) = \frac{0.8 \cdot \frac{}{400}}{0.} =0.002 =0.2\%$

STEP 11

Next, we need to calculate $(B')$ , the total probability of a person testing negative. We can do this using the law of total probability, which states that $(B') =(B'|A)(A) +(B'|A')(A')$ .

STEP 12

We know that $(B'|A) = -(B|A)$ and $(B'|A') = -(B|A')$ . So, plug in the given values for $(B|A)$ and $(B|A')$ to calculate $(B'|A)$ and $(B'|A')$ .
$(B'|A) = -0.8 =0.2$ $(B'|A') = -0. =0.9$

STEP 13

Now, plug in the calculated values for $(B'|A)$ , $(A)$ , $(B'|A')$ , and $(A')$ to calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400}$

STEP 14

Calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400} =0.9$

STEP 15

Finally, we can calculate $(A'|B')$ , the probability of a person not being infected given they have tested negative. We can do this using Bayes' theorem, which states that $(A'|B') = \frac{(B'|A')(A')}{(B')}$ .

STEP 16

Now, plug in the calculated values for $(B'|A')$ , $(A')$ , and $(B')$ to calculate $(A'|B')$ .
$(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9}$

STEP 17

Calculate $(A'|B')$ .
$(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9} =0.9975 =99.75\%$ So, the probability that a person has the virus given that they have tested positive is $0.2\%$ and the probability that a person does not have the virus given that they test negative is $99.75\%$ .

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STEP 1

STEP 2

First, we need to calculate (A′)(A')(A′), the probability of a person not being infected by the virus. We can do this by subtracting (A)(A)(A) from 111.(A′)=1−(A)(A') =1 -(A)(A′)=1−(A)

STEP 3

Now, plug in the given value for (A)(A)(A) to calculate (A′)(A')(A′).(A′)=1−1400(A') =1 - \frac{1}{400}(A′)=1−4001​

STEP 4

Calculate (A′)(A')(A′).(A′)=1−1400=399400(A') =1 - \frac{1}{400} = \frac{399}{400}(A′)=1−4001​=400399​

STEP 5

Next, we need to calculate (B)(B)(B), the total probability of a person testing positive. We can do this using the law of total probability, which states that (B)=(B∣A)(A)+(B∣A′)(A′)(B) =(B|A)(A) +(B|A')(A')(B)=(B∣A)(A)+(B∣A′)(A′).

STEP 6

Now, plug in the given values for (B∣A)(B|A)(B∣A), (A)(A)(A), (B∣A′)(B|A')(B∣A′), and (A′)(A')(A′) to calculate (B)(B)(B).(B)=0.8⋅1400+0.1⋅399400(B) =0.8 \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400}(B)=0.8⋅4001​+0.1⋅400399​

STEP 7

Calculate (B)(B)(B).(B)=0.⋅1400+0.1⋅399400=40400=0.1(B) =0. \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400} = \frac{40}{400} =0.1(B)=0.⋅4001​+0.1⋅400399​=40040​=0.1

STEP 8

Now, we can calculate (A∣B)(A|B)(A∣B), the probability of a person being infected given they have tested positive. We can do this using Bayes' theorem, which states that (A∣B)=(B∣A)(A)(B)(A|B) = \frac{(B|A)(A)}{(B)}(A∣B)=(B)(B∣A)(A)​.

STEP 9

Now, plug in the given values for (B∣A)(B|A)(B∣A), (A)(A)(A), and (B)(B)(B) to calculate (A∣B)(A|B)(A∣B).(A∣B)=.8⋅400.(A|B) = \frac{.8 \cdot \frac{}{400}}{.}(A∣B)=..8⋅400​​

STEP 10

Calculate (A∣B)(A|B)(A∣B).(A∣B)=0.8⋅4000.=0.002=0.2%(A|B) = \frac{0.8 \cdot \frac{}{400}}{0.} =0.002 =0.2\%(A∣B)=0.0.8⋅400​​=0.002=0.2%

STEP 11

Next, we need to calculate (B′)(B')(B′), the total probability of a person testing negative. We can do this using the law of total probability, which states that (B′)=(B′∣A)(A)+(B′∣A′)(A′)(B') =(B'|A)(A) +(B'|A')(A')(B′)=(B′∣A)(A)+(B′∣A′)(A′).

STEP 12

STEP 13

Now, plug in the calculated values for (B′∣A)(B'|A)(B′∣A), (A)(A)(A), (B′∣A′)(B'|A')(B′∣A′), and (A′)(A')(A′) to calculate (B′)(B')(B′).(B′)=0.2⋅400+0.9⋅399400(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400}(B′)=0.2⋅400​+0.9⋅400399​

STEP 14

Calculate (B′)(B')(B′).(B′)=0.2⋅400+0.9⋅399400=0.9(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400} =0.9(B′)=0.2⋅400​+0.9⋅400399​=0.9

STEP 15

STEP 16

Now, plug in the calculated values for (B′∣A′)(B'|A')(B′∣A′), (A′)(A')(A′), and (B′)(B')(B′) to calculate (A′∣B′)(A'|B')(A′∣B′).(A′∣B′)=0.9⋅3994000.9(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9}(A′∣B′)=0.90.9⋅400399​​

STEP 17

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First, we need to calculate $(A')$ , the probability of a person not being infected by the virus. We can do this by subtracting $(A)$ from $1$ .
$(A') =1 -(A)$

Now, plug in the given value for $(A)$ to calculate $(A')$ .
$(A') =1 - \frac{1}{400}$

Calculate $(A')$ .
$(A') =1 - \frac{1}{400} = \frac{399}{400}$

Next, we need to calculate $(B)$ , the total probability of a person testing positive. We can do this using the law of total probability, which states that $(B) =(B|A)(A) +(B|A')(A')$ .

Now, plug in the given values for $(B|A)$ , $(A)$ , $(B|A')$ , and $(A')$ to calculate $(B)$ .
$(B) =0.8 \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400}$

Calculate $(B)$ .
$(B) =0. \cdot \frac{1}{400} +0.1 \cdot \frac{399}{400} = \frac{40}{400} =0.1$

Now, we can calculate $(A|B)$ , the probability of a person being infected given they have tested positive. We can do this using Bayes' theorem, which states that $(A|B) = \frac{(B|A)(A)}{(B)}$ .

Now, plug in the given values for $(B|A)$ , $(A)$ , and $(B)$ to calculate $(A|B)$ .
$(A|B) = \frac{.8 \cdot \frac{}{400}}{.}$

Calculate $(A|B)$ .
$(A|B) = \frac{0.8 \cdot \frac{}{400}}{0.} =0.002 =0.2\%$

Next, we need to calculate $(B')$ , the total probability of a person testing negative. We can do this using the law of total probability, which states that $(B') =(B'|A)(A) +(B'|A')(A')$ .

Now, plug in the calculated values for $(B'|A)$ , $(A)$ , $(B'|A')$ , and $(A')$ to calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400}$

Calculate $(B')$ .
$(B') =0.2 \cdot \frac{}{400} +0.9 \cdot \frac{399}{400} =0.9$

Now, plug in the calculated values for $(B'|A')$ , $(A')$ , and $(B')$ to calculate $(A'|B')$ .
$(A'|B') = \frac{0.9 \cdot \frac{399}{400}}{0.9}$