Awww, I wanted to explain it... and sound really smug and uppity while doing so...(!) =PLlil said:
MATH questions - It has begun once again!
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You can't do that.BiscuitTrouser said:
2[sup]x/2[/sup]/2 = 2[sup]x/2 - 1[/sup] not 1[sup]x/2[/sup]
Remember, if you're not sure if a method is valid or not, you can always check using something you do know.
For example, take the equation 2[sup]x[/sup] = 8, then x obviously equals 3, but lets try and solve it with your method. In that case we get 2[sup]x[/sup]/2 = 8/2 which then goes to 1[sup]x[/sup] = 4. But 1 to the power of anything is 1, so we get 1 = 4. From that we can deduce something has gone wrong.
As for the question at hand, this [http://www.newton.dep.anl.gov/askasci/math99/math99274.htm] seems to be the best explanation of the situation I can find quickly.
Here's a simple one, with just some wordplay to work through:
You have a book with 200 pages. The first half is half-filled with words; of the rest, half of the pages are half-truths, and less than half of them half as scrupulous as you would like. What page number would you be on if you went half-way through the second half of the book, then back half the length of the book, continued reading half of half of half of the book, read half the product of half the total pages and one-half, and retreated another half the length of the book, then counted half that many pages from the very end of the book?
EDIT:
Reworded to make sense... sort of.
You have a book with 200 pages. The first half is half-filled with words; of the rest, half of the pages are half-truths, and less than half of them half as scrupulous as you would like. What page number would you be on if you went half-way through the second half of the book, then back half the length of the book, continued reading half of half of half of the book, read half the product of half the total pages and one-half, and retreated another half the length of the book, then counted half that many pages from the very end of the book?
EDIT:
Reworded to make sense... sort of.
Ok, I am going to explain the answer to my question. I am going to go pretty far into detail.
Theoretically, this is a pretty simple algebra problem. Anyone who has passed a course in algebra should be able to answer this question correctly. But I generally find most high school graduates (who have not yet completed a math course in college) don't get part A right. 438 is the most common answer (I used to tutor to help people catch up in algebra before they went to college. I usually don't include part B and C.) When people answer it wrong it is because they are not really reading the problem, which is a common problem with . They see a bunch of numbers and start multiplying and subtracting and whatnot without really considering all the information given within a question or what is being asked.
We are told that "if one has the disease, the test comes back "yes" 98% of the time (and "no" 2% of the time), and if one does not have the disease, the test comes back "no" 98% of the time (and "yes" 2% of the time)." This is where most of you went wrong.
What we know from this statement:
1. 2% of the people who have the disease will test negative.
2. 2% of the people who do not have the disease will test positive.
What most of you based your calculations on:
3. 2% of the people who tested positive do not have the disease.
4. 2% of the people who tested negative do have the disease.
This is a logic error. It does not hold that because 2% of the tests are failures that 2% of each result category are incorrect.
"You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability." So, here is all the information we have:
1. 2% of the people who have the disease will test negative.
2. 2% of the people who do not have the disease will test positive.
3. 10000 total population
4. 248 tested positive.
5. Reality exactly matches probability in this case.
Now, the question (part A)
"A) How many people are actually sick?"
Basically, Find x where x is the amount of truly sick people.
x = truly sick
One more variable so this will be easier to understand.
y = Healthy people
Now, let's make some equations:
Total Population - sick people = healthy people (this should be obvious)
10000 - x = Y
Next, lets explain the test results in an equation:
2% of Healthy people + 98% of sick people = total positive results. (We know this from information points 1 and 2. We also have a value for "total positive results" from information point 4. We know this equation is accurate because of information point 5.)
0.02Y + 0.98x = 248
or, to make it more usable,
y/50 + 49x/50 = 248
What we have here is a simple multiple variable problem. We need to solve for x, so substitute out y. From our first equation, we already know the value of y.
(10000 - x)/50 + 49x/50 = 248
Solve for X.
10000 - x + 49x = 12400
48X = 2400
x = 50
There are 50 actually sick people.
"B) You tested negative. What is the chance you are sick?"
What we need here is how many people who tested negative are actually sick. For this, we need how many people tested negative total (10000 - 248 = 9752) and How many people who were sick tested negative (50 * 0.02 = 1) So, 9752 people tested negative and 1 of them is actually sick.
1/9752 chance or about 0.01% chance.
"C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?"
Opposite of the other problem. How many people who tested positive are actually healthy? we need total positive (248) and how many people who are not sick tested positive ((10000 - 50)*0.02 = 199) So, 199 of 248 are healthy.
199/248 chance or about 80.24% chance.
What we know from this statement:
1. 2% of the people who have the disease will test negative.
2. 2% of the people who do not have the disease will test positive.
What most of you based your calculations on:
3. 2% of the people who tested positive do not have the disease.
4. 2% of the people who tested negative do have the disease.
This is a logic error. It does not hold that because 2% of the tests are failures that 2% of each result category are incorrect.
"You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability." So, here is all the information we have:
1. 2% of the people who have the disease will test negative.
2. 2% of the people who do not have the disease will test positive.
3. 10000 total population
4. 248 tested positive.
5. Reality exactly matches probability in this case.
Now, the question (part A)
"A) How many people are actually sick?"
Basically, Find x where x is the amount of truly sick people.
x = truly sick
One more variable so this will be easier to understand.
y = Healthy people
Now, let's make some equations:
Total Population - sick people = healthy people (this should be obvious)
10000 - x = Y
Next, lets explain the test results in an equation:
2% of Healthy people + 98% of sick people = total positive results. (We know this from information points 1 and 2. We also have a value for "total positive results" from information point 4. We know this equation is accurate because of information point 5.)
0.02Y + 0.98x = 248
or, to make it more usable,
y/50 + 49x/50 = 248
What we have here is a simple multiple variable problem. We need to solve for x, so substitute out y. From our first equation, we already know the value of y.
(10000 - x)/50 + 49x/50 = 248
Solve for X.
10000 - x + 49x = 12400
48X = 2400
x = 50
There are 50 actually sick people.
"B) You tested negative. What is the chance you are sick?"
What we need here is how many people who tested negative are actually sick. For this, we need how many people tested negative total (10000 - 248 = 9752) and How many people who were sick tested negative (50 * 0.02 = 1) So, 9752 people tested negative and 1 of them is actually sick.
1/9752 chance or about 0.01% chance.
"C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?"
Opposite of the other problem. How many people who tested positive are actually healthy? we need total positive (248) and how many people who are not sick tested positive ((10000 - 50)*0.02 = 199) So, 199 of 248 are healthy.
199/248 chance or about 80.24% chance.
A) There are 50 actually sick people.
B) 1/9752 chance or about 0.01% chance.
C) 199/248 chance or about 80.24% chance.
B) 1/9752 chance or about 0.01% chance.
C) 199/248 chance or about 80.24% chance.