A new test has been developed to detect a particular type of cancer. The test must be evaluated before it is put into use. A med
ical researcher selects a random sample of 1,000 adults and finds (by other means) that 4% have this type of cancer. Each of the 1,000 adults is given the new test, and it is found that the test indicates cancer in 99% of those who have it and in 1% of those who do not.
a) Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b) What is the probability of a person having cancer given that the test does not indicate cancer?
a) Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b) What is the probability of a person having cancer given that the test does not indicate cancer?
1 answer:
The tree diagram of the problem above is attached
There are four outcomes of the two events,
First test - Cancer, Second Test - Cancer, the probability is 0.0396
First test - Cancer, Second Test - No Cancer, the probability is 0.0004
First test - No Cancer, Second Test - There is cancer, the probability is 0.0096
First test - No cancer, Second Test - No cancer, the probability is 0.9054
The probability of someone picked at random has cancer given that test result indicates cancer is
The probability of someone picked at random has cancer given that test result indicates no cancer is
There are four outcomes of the two events,
First test - Cancer, Second Test - Cancer, the probability is 0.0396
First test - Cancer, Second Test - No Cancer, the probability is 0.0004
First test - No Cancer, Second Test - There is cancer, the probability is 0.0096
First test - No cancer, Second Test - No cancer, the probability is 0.9054
The probability of someone picked at random has cancer given that test result indicates cancer is
The probability of someone picked at random has cancer given that test result indicates no cancer is
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The area of the shaded region /ACDEH/ is 325.64cm²
Step 1 - Collect all the facts
First, let's examine all that we know.
- We know that the octagon is regular which means all sides are equal.
- since all sides are equal, then all sides are equal 10cm.
- if all sides are equal then all angles within it are equal.
- since the total angle in an octagon is 1080°, the sum of each angle within the octagon is 135°.
Please note that the shaded region comprises a rectangle /ADEH/ and a scalene triangle /ACD/.
So to get the area of the entire region, we have to solve for the area of the Scalene Triangle /ACD/ and add that to the area of the rectangle /ADEH/
Step 2 - Solving for /ACD/
The formula for the area of a Scalene Triangle is given as:
A =
This formula assumes that we have all the sides. But we don't yet.
However, we know the side /CD/ is 10cm. Recall that side /CD/ is one of the sides of the octagon ABCDEFGH.
This is not enough. To get sides /AC/ and /AD/ of Δ ACD, we have to turn to another triangle - Triangle ABC. Fortunately, ΔABC is an Isosceles triangle.
Step 3 - Solving for side AC.
Since all the angles in the octagon are equal, ∠ABC = 135°.
Recall that the total angle in a triangle is 180°. Since Δ ABC is an Isosceles triangle, sides /AB/ and /BC/ are equal.
Recall that the Base angles of an isosceles triangle is always equal. That is ∠BCA and ∠BAC are equal. To get that we say:
180° - 135° = 45° [This is the sum total of ∠BCA and ∠BAC. Each angle therefore equals
45°/2 = 22.5°
Now that we know all the angles of Δ ABC and two sides /AB/ and /BC/, let's try to solve for /AC/ which is one of the sides of Δ ACD.
According to the Sine rule,
=
=
Since we know side /BC/, let's go with the first two parts of the equation.
That gives us
Cross multiplying the above, we get
/AC/ =
Side /AC/ = 18.48cm.
Returning to our Scalene Triangle, we now have /AC/ and /CD/.
To get /AD/ we can also use the Sine rule since we can now derive the angles in Δ ABC.
From the Octagon the total angle inside /HAB/ is 135°. We know that ∠HAB comprises ∠CAB which is 22.5°, ∠HAD which is 90°. Therefore, ∠DAC = 135° - (22.5+90)
∠DAC = 22.5°
Using the same deductive principle, we can obtain all the other angles within Δ ACD, with ∠CDA = 45° and ∠112.5°.
Now that we have two sides of ΔACD and all its angles, let's solve for side /AD/ using the Sine rule.
=
Cross multiplying we have:
/AD/ =
Therefore, /AD/ = 24.15cm.
Step 4 - Solving for Area of ΔACD
Now that we have all the sides of ΔACD, let's solve for its area.
Recall that the area of a Scalene Triangle using Heron's formula is given as
A =
Where S is the semi-perimeter given as
S= (/AC/ + /CD/ + /DA/)/2
We are using this formula because we don't have the height for ΔACD but we have all the sides.
Step 5 - Solving for Semi Perimeter
S = (18.48 + 10 + 24.15)/2
S = 26.32
Therefore, Area =
A =
A = Square cm.
A of ΔACD = 85.49cm²
Recall that the shape consists of the rectangle /ADEH/.
The A of a rectangle is L x B
A of /ADEH/ = 240.15cm²
Step 6 - Solving for total Area of the shaded region of the Octagon
The total area of the Shaded region /ACDEH/, therefore, is 240.15 + 85.49
= 325.64cm²
See the link below for more about Octagons:
brainly.com/question/4515567
