MATH questions - It has begun once again!

[DrOswald]

Suppose that we have a test for a disease that is 98% accurate: 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). You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability.

A) How many people are actually sick?

B) You tested negative. What is the chance you are sick?

C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?

 

[DrOswald]

9 + 2 + 2 ? That would mean the sum is 13. But what does that matter? Incidentally, 13 * 3 = 36. But I still don't see it.

Bob knows the house number, so that would mean he knows the sum, right? That's important too.

Bob knows the house number, yes. But I do not. And I am suppose to imagine being Bob in that riddle. If I knew the neighbors house number was 13, then I would have known there would be only 1 solution (9, 2, 2). But how am I suppose to know the house number is 13?

Meh, it is probably so simple I can't even think of it. XD

Bob needs more info because there are 2 possibilities for the number he saw. There is only 1 number (13) that has 2 solutions (2+2+9 and 1+6+6.) If we assume that "my oldest plays piano" means that the oldest child is not part of a twin pair, then there is only 1 solution remaining.

 

[Llil]

Bob knows the house number, yes. But I do not. And I am suppose to imagine being Bob in that riddle. If I knew the neighbors house number was 13, then I would have known there would be only 1 solution (9, 2, 2). But how am I suppose to know the house number is 13?

Meh, it is probably so simple I can't even think of it. XD

Try writing out the sums of all possible combinations (there's 8). Bob knows the number, but there's also the piano part. Now, why's that necessary?

EDIT - And the solution is in the post above.

 

[4RM3D]

Bob needs more info because there are 2 possibilities for the number he saw. There is only 1 number (13) that has 2 solutions (2+2+9 and 1+6+6.) If we assume that "my oldest plays piano" means that the oldest child is not part of a twin pair, then there is only 1 solution remaining.

Try writing out the sums of all possible combinations (there's 8). Bob knows the number, but there's also the piano part. Now, why's that necessary?

EDIT - And the solution is in the post above.

There are 8 possibilities? I can only find 7:
1 * 2 * 18 = 36
1 * 3 * 12 = 36
1 * 4 * 9 = 36
1 * 6 * 6 = 36
2 * 2 * 9 = 36
2 * 3 * 6 = 36
3 * 3 * 4 = 36

I already mentioned, I know the reason about why you needed to mention there is an oldest child.

And I know if you need to have a sum of 13 that there is only 1 solution left.

But...

Where does the number 13 come from? How do I know that the neighbors house is 13?

 

[isometry]

Suppose that we have a test for a disease that is 98% accurate: 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). You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability.

A) How many people are actually sick?

B) You tested negative. What is the chance you are sick?

C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?

A) Let n be the number of sick people. Then the expected number of tests that return "yes" is:

expected sick population = (10000 - n)*0.02 + 0.98 n

You state that the sick population is 248, so solving the equation gives n = 50 people who are actually sick.

B) The quantity we want is P(sick | "neg"), the probability that I'm sick given the answer I got is "negative." Using Bayes theorem:

P(sick | "neg") = P("neg" | sick) * P(sick)/P("neg")

We are told that P("neg" | sick) = 0.02. We found P(sick) = 50/10000 = 0.005 in part (A), and we know P("neg") = 248/10000 from the premise of the problem. Putting these together we get:

P(sick | "no") = 0.02*50/248 = 0.004.

C) Similar to part (B), again we use Bayes' theorem:

P(healthy | "pos") = P("pos" | healthy) * P(healthy)/P("pos") = 0.02*9550/9752 = 0.02.

 

[Melon Hunter]

Suppose that we have a test for a disease that is 98% accurate: 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). You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability.

A) How many people are actually sick?

B) You tested negative. What is the chance you are sick?

C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?

A) 248 * 0.98 + (10000-248) * 0.02 = 438 sick people (rounded down from 438.08)

B) and C) are both 0.02 by the test's given accuracy. C) is simply the inverse situation of B), so they have the same answer.

 

[The Unworthy Gentleman]

cosx. I know that as a rule.

A man drops a ball off of a building. It takes 10 seconds to hit the ground. The ball weighs basically nothing and there is no wind, the ball generates negligable air resistance. How tall is the building.

I expect this is a trap, but I'm going to say 490 metres; it seems too simple at first glance.

OT: 2[sup]x[/sup] = x[sup]2[/sup]

Solve without using observation. I don't know the answer so you're trying to convince me you're right, don't forget the commentary of what you're doing. People who aren't too good at maths need to know what's going on.

 

[Maze1125]

P(A) = the probability both kids are boys = 1/4

P(B) = the probability one of the kids is a boy = 1/2

P(B|A) = the probability that one kid is a boy, given that both are boys = 100%

Plug this into Bayes theorem and we get:

P(A|B) = (1.0)*(.25)/(0.5) = 1/2 = 50%

So, the probability that both kids are boys, given that one of them is a boy, is 50%. The "cross out possibilities that don't happen and split the probability equally on the others" approach only works if the events are independent, and in this case they are not.

The bit in bold is wrong.

If event B is "At least one of the children is a boy." then P(B) = 3/4 not 1/2.

Or, if event B is "Exactly one of the children is a boy." then P(B) = 1/2, but then P(B|A) = P(A|B) = 0

If you mean the former, as I suspect given the question at hand, then Bayes' Theorem gives P(A|B) = (1/4)/(3/4) = 1/3

Incidentally, to the others, the time it takes to fall "t = 10" is given to only one significant digit,

You don't know that, the significant figures aren't stated and two digits are shown.

So it could either be that case that the real figure is something like 12.3 given to 1 s.f. or 10.3 given to 2 s.f.

Just because the last digit is 0 doesn't mean it's been made zero due to rounding.

 

[4RM3D]

Suppose that we have a test for a disease that is 98% accurate: 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). You are in a population of 10,000 people, 248 of which test positive. Assume reality exactly matches probability.

A) How many people are actually sick?

B) You tested negative. What is the chance you are sick?

C) They sent you the wrong envelope, you actually tested positive. What is the chance you are healthy?

A) (248 * 98%) + ((10.000 - 248) * 2%)

B) 2% chance you are sick?

C) 2% chance you are healthy?

EDIT: and it seems a few other people beat me to the answer.

 

[Llil]

There are 8 possibilities? I can only find 7
...
Where does the number 13 come from? How do I know that the neighbors house is 13?

There's also 1 * 1 * 36
If you write down the sums, you'll get
1 + 1 + 36 = 38
1 + 2 + 18 = 21
1 + 3 + 12 = 16
1 + 4 + 9 = 14
1 + 6 + 6 = 13
2 + 2 + 9 = 13
2 + 3 + 6 = 11
3 + 3 + 4 = 10

And even after seeing the house number, Bob came back for another clue, so...

 

[BiscuitTrouser]

cosx. I know that as a rule.

A man drops a ball off of a building. It takes 10 seconds to hit the ground. The ball weighs basically nothing and there is no wind, the ball generates negligable air resistance. How tall is the building.

I expect this is a trap, but I'm going to say 490 metres; it seems too simple at first glance.

OT: 2[sup]x[/sup] = x[sup]2[/sup]

Solve without using observation. I don't know the answer so you're trying to convince me you're right, don't forget the commentary of what you're doing. People who aren't too good at maths need to know what's going on.

Well one obvious solution is 2.

When x = 2

As 2[sup]2[/sup] = 2[sup]2[/sup] it obviously cancles. As for without observation?

Lets give it a crack. First lets square root both sides. Squre root is the same as putting everything to the power of 1/2. If you put something thats already to a power to another power on top you multply them. This means (2^x)^1/2 = 2^1/2x.

We then get 2[sup]1/2x[/sup] = x

Thats a start. Not sure if this is a cheat but lets divide by 2.

Then we get 1[sup]1/2x[/sup] = x/2

1 power anything is one. So we get 1 = x/2 so x = 2. Not sure if that counts?

 

[4RM3D]

There are 8 possibilities? I can only find 7
...
Where does the number 13 come from? How do I know that the neighbors house is 13?

There's also 1 * 1 * 36
If you write down the sums, you'll get
1 + 1 + 36 = 38
1 + 2 + 18 = 21
1 + 3 + 12 = 16
1 + 4 + 9 = 14
1 + 6 + 6 = 13
2 + 2 + 9 = 13
2 + 3 + 6 = 11
3 + 3 + 4 = 10

And even after seeing the house number, Bob came back for another clue, so...

And there is the kicker. Nope, wouldn't have guessed it. Well, maybe if I wasn't trying to solve problems in the middle of the night, while I actually need to go to bed and sleep.

Anyhow, thanks for explaining.

 

[isometry]

P(A) = the probability both kids are boys = 1/4

P(B) = the probability one of the kids is a boy = 1/2

P(B|A) = the probability that one kid is a boy, given that both are boys = 100%

Plug this into Bayes theorem and we get:

P(A|B) = (1.0)*(.25)/(0.5) = 1/2 = 50%

So, the probability that both kids are boys, given that one of them is a boy, is 50%. The "cross out possibilities that don't happen and split the probability equally on the others" approach only works if the events are independent, and in this case they are not.

The bit in bold is wrong.

If event B is "At least one of the children is a boy." then P(B) = 3/4 not 1/2.

Or, if event B is "Exactly one of the children is a boy." then P(B) = 1/2, but then P(B|A) = P(A|B) = 0

If you mean the former, as I suspect given the question at hand, then Bayes' Theorem gives P(A|B) = (1/4)/(3/4) = 1/3

You're right. Maybe one day I will learn to do simple probability problems correctly, but not today, damn.

Incidentally, to the others, the time it takes to fall "t = 10" is given to only one significant digit,

You don't know that, the significant figures aren't stated and two digits are shown.

So it could either be that case that the real figure is something like 12.3 given to 1 s.f. or 10.3 given to 2 s.f.

Just because the last digit is 0 doesn't mean it's been made zero due to rounding.

Depends on the convention, I would say the significant digits are shown, that if there were two significant digits we should have been given the duration of the fall as 1.0 * 10^1. I'll retract my statement that the answer "500" is more correct, but at the very least it's equally correct.

 

[omega 616]

EDIT: formula --> 5 * TAN(((90 - 60) * Pi) / 180)

No, it's the tan of the angle multiplied by the adjacent.
So it's just 5 * tan(60)

Spoiler: so it's roughly

8.66 metres

Meanwhile, I'm struggling with the pyramid problem :/

Is it wrong I was never taught trigonometry or differentiation and yet I

Spoiler: guessed

9

?

Shit I was never even taught times tables, no seriously. I hardcore suck at maths, luckily differentiation and trigonometry are never used in life and everything has a calculator on it these days.

 

[The Unworthy Gentleman]

Well one obvious solution is 2.

When x = 2

As 2[sup]2[/sup] = 2[sup]2[/sup] it obviously cancles. As for without observation?

Lets give it a crack. First lets square root both sides. Squre root is the same as putting everything to the power of 1/2. If you put something thats already to a power to another power on top you multply them. This means (2^x)^1/2 = 2^1/2x.

We then get 2[sup]1/2x[/sup] = x

Thats a start. Not sure if this is a cheat but lets divide by 2.

Then we get 1[sup]1/2x[/sup] = x/2

1 power anything is one. So we get 1 = x/2 so x = 2. Not sure if that counts?

You reach a legitimate answer, but you used observation when you got to 1[sup]1/2x[/sup] = 1.

 

[Blade1130]

The much simpler, actually sort of Computer Science question rather than math would be:

What is 5 % 3?

Spoiler: Answer

2

The % (modulus) operator takes the first number and divides it by the second number, then returns the remainder. 5 / 3 = 1 R: 2 then we forget about the 1, and 2 is the answer.

Basically, a % b = The remainder when a / b.

If you've never heard of the modulus operator, you probably weren't taught it. It's completely useless in math, but has certain applications in Computer Science. Although I did use it on the Calculus test once. We needed a formula that would give you the nth derivative of a function, but it kept switching between + and -. Now I realize you just need to do (-1)^n, but at the time I didn't realize that and instead did something along the lines of ((n % 2) * 2) - 1. It gives you either -1 or 1 and switches every time.

Calc teacher wrote "Don't make stuff up" on the test, though my Comp Sci teacher said if it were his test he would've given me comedy points.

The confusing as crap question that everyone will be pissed at me for because of the terrible answer would be this: I was working on a video game in which the player would shoot a bullet, and the bullet would bounce off of a circle in the middle of the screen.

In essence, if you know the equation of the circle (Or rather, both, since a circle is technically two functions), the point of contact, and the vector of the bullet immediately before it hits, how do you find the resulting vector it should bounce?

Spoiler: Half-assed answer

I haven't condensed this down into a single formula (it'd be too difficult to read in one line anyways), but the way I did it was finding the tangent line to the circle at the point of contact by taking a derivative of the circle at that X value, then using the point-slope formula to determine the Y-intercept. At that point I assumed that the angle between the incoming vector and line would be the same as the angle between the outgoing vector and the line. Originally, I tried using inverse tangents and subtracting the angles, which does work mathematically, if that's how you did it, but I couldn't get the computer to do it, since there's no way to really know what quadrant it's in and computers are really dumb. I ended up taking the line perpendicular to the tangent, passing through the point of contact, and then reflected the vector over that line. Since a vector is really just 2 points connected with direction, and the first point was at the point of contact, while the second was simply that same point minus the incoming vector. I found the distance between the second point and the line perpendicular to the tangent, then just went that same distance on the other side of the line, put a new point there and connected it to the point of contact, giving you the resulting vector.

Hard to explain without a picture, sooner or later I'll put it up on my website, if anyone wants I can link then later. It has a nice debugging mode that draws the tangent line and the points I'm mentioning.

 

[gandhi the peacemake]

You got a product that costs $100. How much does the product cost without 20% tax (VAT). (Yes, tax is not the same in every country. But for the sake of this question the tax is 20%)

A better way to phrase this question would be:

"You buy something that TOTALS $100 after a 20% tax. How much did the item cost before tax?" But that's just the English major in me making an ass of myself.

You have 2 separate events (triggers) that can happen simultaneously. Each event has a 50% chance of happening. What is the chance that event 1 or event 2 or both happens?

Spoiler

Possible outcomes:

Chance of only event 1 occurring: 25%
Chance of only event 2 occurring: 25%
Chance of both events occurring: 25%
Chance of neither events occurring: 25%

The chance of only event 1 occurring OR only event 2 occurring OR both events occurring = 75%

As for the 4m ladder/5m ladder question,

Spoiler

I got the same 1.973m, but Christ I didn't use as heavy math skills as you (partly because I didn't need to, and partly because I don't possess them durr me).

Lacking a diagram, I hope I can explain this just in words, and I'll try to be as simple as possible.

Draw two right triangles (to represent the triangles the two ladders create in the alley).
Their dimensions:
- 2 x Y(1) x 4
- 2 x Y(2) x 5

Using ol' sohcahtoa:
- the base angle of the 2 x Y(1) x 4 triangle = 60 degrees
- the base angle of the 2 x Y(2) x 5 triangle = 66.422 degrees

Now draw your attention to the triangle formed by the intersection of the two. Split said triangle into two more by way of h, where h is the height. The bases of these two triangles then become (x) and (2-x), respectively.

The height h can be calculated thusly:
- tan(60) = h / (2 - x)
- tan(66.422) = h / x

Simplify the first equation, and solve for h:
- tan(60) = h / (2 - x)
- 1.732 = h / (2 - x)
- 1.732 (2 - x) = h

Simplify the second equation, and solve for h:
- tan(66.422) = h / x
- 2.291 = h / x
- 2.291x = h

Substitute and solve for x:
- 1.732 (2 - x) = 2.291x
- 3.464 - 1.732x = 2.291x
- 3.464 = 4.023x
- 0.861m = x

Plug x back to solve for h:
- tan(66.422) = h / x
- 2.291 = h / 0.861
- 1.973m = h

So the height h, or where the ladders meet, is 1.973 meters.

...or, y'know, stuff. Cool thread; I'd forgotten how much fun these math problems could be. Keep it up, fellas. I'm looking forward to seeing more.

EDIT:

For the sake of newcomers, would it be possible for the topic starter to simply add the questions to the first post, along with an update time/date so we can keep track of those we've already worked on?

 

[BiscuitTrouser]

Well one obvious solution is 2.

When x = 2

As 2[sup]2[/sup] = 2[sup]2[/sup] it obviously cancles. As for without observation?

Lets give it a crack. First lets square root both sides. Squre root is the same as putting everything to the power of 1/2. If you put something thats already to a power to another power on top you multply them. This means (2^x)^1/2 = 2^1/2x.

We then get 2[sup]1/2x[/sup] = x

Thats a start. Not sure if this is a cheat but lets divide by 2.

Then we get 1[sup]1/2x[/sup] = x/2

1 power anything is one. So we get 1 = x/2 so x = 2. Not sure if that counts?

You reach a legitimate answer, but you used observation when you got to 1[sup]1/2x[/sup] = 1.

Not really, 1 to the power of anything is one, i didnt observe a value of X, i just know that for ALL values of X the right hand side equated to one andas such i could replace the right hand side with one. 1^1/2x is the same as 1^1000000 or 1^-175 and it still equals one. It wasnt a question of seeing where x was definately one, it was were in all instances 1 to the power of anything was 1 so the X was irrelivant and could be removed as a variable.