Anyone here into math?

[Guest [Kp...]]

the partial sum of the series 1+2+3+4...+n = n(n+1)/2

 

let us call this sum S.

 

S is also called the triangular number because if the numbers are arranged one below the other, they blow up as an equilateral triangle.

 

can you use the area of the equilateral triangle (or any shape you might wish to arrange the sequence in) to compute the sum of the series, or, S? in other words, show that the formula for the area of the shape (say, equilateral triangle) divided by the area of each number (1 unit sq. if each number is represented by a square of unit size 1) is n(n+1)/2, or, S?

 

(this is Part 1 of the question. Part 2 will follow once Part 1 has been attempted.)

[Guest [Kp...]]

a. Roll a 6-sided die 4 times. You win if you roll a 6.

b. Roll 2 6-sided dice 24 times. You win if you roll double 6s.

 

a

 

probability of not (6) on one roll = 5/6

probability of not (6) in 4 rolls = (5/6)^4 = 0.48

probability of 6 in 4 rolls = 1 - 0.48 = 0.52

 

b

 

probability of 66 on one roll = 1/(6*6) = 1/36

probability of not (66) on one roll = 35/36

probability of not (66) on 24 rolls  = (35/36)^24 = 0.51

probability of 66 on 24 rolls = 1 - 0.51 = 0.49

 

so a.

 

is this right?

 

[[ma...]]

Yes, I get the same

[Guest [ou...]]

a. Roll a 6-sided die 4 times. You win if you roll a 6.

b. Roll 2 6-sided dice 24 times. You win if you roll double 6s.

 

a

 

probability of not (6) on one roll = 5/6

probability of not (6) in 4 rolls = (5/6)^4 = 0.48

probability of 6 in 4 rolls = 1 - 0.48 = 0.52

 

b

 

probability of 66 on one roll = 1/(6*6) = 1/36

probability of not (66) on one roll = 35/36

probability of not (66) on 24 rolls  = (35/36)^24 = 0.51

probability of 66 on 24 rolls = 1 - 0.51 = 0.49

 

so a.

 

is this right?

 

Exactly!

[Guest [ou...]]

the partial sum of the series 1+2+3+4...+n = n(n+1)/2

 

let us call this sum S.

 

S is also called the triangular number because if the numbers are arranged one below the other, they blow up as an equilateral triangle.

 

can you use the area of the equilateral triangle (or any shape you might wish to arrange the sequence in) to compute the sum of the series, or, S? in other words, show that the formula for the area of the shape (say, equilateral triangle) divided by the area of each number (1 unit sq. if each number is represented by a square of unit size 1) is n(n+1)/2, or, S?

 

Let us call the area of an "equilateral triangle" of N dots T(N).

2 copies of such a triangle form an Nx(N+1) rectangle.

It follows then that T(N)=N*(N+1)/2.

Q.E.D.

[Guest [ou...]]

i read the following about "average velocity" in the wikipedia article and again got confused (because it is contradictory to what you say):

 

You can only talk about velocity at a given moment (instant).

 

If you go from A to B and back your displacement is 0,

which would make your velocity 0, which is absurd.

You can always talk about your average velocity which is (AB)/t(A->B).

 

The big difference can be noticed when we consider movement around a circle. When something moves in a circular path (at a constant speed, see above) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle.

 

There is no contradiction if you substitute 'velocity' for 'speed'.

In Greek we use the same word for both velocity and speed: tachytis (as in tachycardia).

In English velocity and speed can't always be used interchangeably.

 

Due to the facts that:

1. I'm Greek,

2. average velocity and speed are both scalar, and

3. oftentimes equal (sometimes average velocity < average speed)

I used the term velocity instead of speed, which in turn caused your confusion.

 

i was wondering if i could ask you to explain to me some old math concepts that i never understood?

 

Of course you can.

[Guest [Kp...]]

the partial sum of the series 1+2+3+4...+n = n(n+1)/2

 

let us call this sum S.

 

S is also called the triangular number because if the numbers are arranged one below the other, they blow up as an equilateral triangle.

 

can you use the area of the equilateral triangle (or any shape you might wish to arrange the sequence in) to compute the sum of the series, or, S? in other words, show that the formula for the area of the shape (say, equilateral triangle) divided by the area of each number (1 unit sq. if each number is represented by a square of unit size 1) is n(n+1)/2, or, S?

 

Let us call the area of an "equilateral triangle" of N dots T(N).

2 copies of such a triangle form an Nx(N+1) rectangle.

It follows then that T(N)=N*(N+1)/2.

Q.E.D.

 

very very neat! i know this is right but i am having trouble creating a rectangle of two equilateral triangles. i'm going to think about this. if i fail, i'll edit this post and ask for more info.

 

i solved it using a 60 degree rhombus to represent a number and tiled accordingly (see image):

 

 

i was wondering if i could ask you to explain to me some old math concepts that i never understood?

 

Of course you can.

 

thank you sir -- kind of you.

 

i was contacted by colin and pianogirl yesterday after my confessional. i might have to leave again -- i don't know. please give me a few days.

[Guest [ou...]]

I see no proposals so let's recap first.

 

 

Solved Problems:

 

1. A car travels a given distance going 90km/h for the 1st half and 110km/h for the 2nd.

What's the average speed of the car?

 

The answer is the harmonic mean V=2/(1/90+1/110)=99km/h.

 

 

2. Which bet would you take and why:

A. Roll a 6-sided die 4 times. You win if you roll a 6.

B. Roll 2 6-sided dice 24 times. You win if you roll double 6s.

 

P(A)=1-P(A')=1-(5/6)^4=0.518=51.8%

P(B)=1-P(B')=1-(35/36)^24=0.491=49.1%

 

 

Proposed Problems:

 

1. A family has 2 kids. 1 of them is a girl.

What's the probability of the other kid being also a girl?

 

2. Given an event A, find the minimum of 1/P(A)+1/P(A').

 

3. In a game of Russian roulette, which player should go 1st and why?

 

Edit: I realized that the wording of P3 was poor. I'll be reposting it properly.

[Guest [Kp...]]

1. A family has 2 kids. 1 of them is a girl.

What's the probability of the other kid being also a girl?

 

outcomes: GG, BG, GB, BB

only first three comply necessary condition (one of them is a girl), thus outcomes: GG, BG, GB

 

probability of GG = 1/3

[Guest [Kp...]]

3. In a game of Russian roulette, which player should go 1st and why?

 

edited...

[Guest [ou...]]

1. A family has 2 kids. 1 of them is a girl.

What's the probability of the other kid being also a girl?

 

outcomes: GG, BG, GB, BB

only first three comply necessary condition (one of them is a girl), thus outcomes: GG, BG, GB

 

probability of GG = 1/3

 

Didn't fall in the 50% trap. Nice!

 

3. In a game of Russian roulette, which player should go 1st and why?

 

edited...

 

??

[Guest [Kp...]]

1. A family has 2 kids. 1 of them is a girl.

What's the probability of the other kid being also a girl?

 

outcomes: GG, BG, GB, BB

only first three comply necessary condition (one of them is a girl), thus outcomes: GG, BG, GB

 

probability of GG = 1/3

 

Didn't fall in the 50% trap. Nice!

 

i've learnt the hard way that if you have a choice, it is safer to enumerate the events. 

 

3. In a game of Russian roulette, which player should go 1st and why?

 

edited...

 

??

 

i was thinking it is like the monty hall problem but i realized we do not have the option of switching chambers after the first shot! intuitively my answer then is "it does not matter who goes first" but i know it is wrong (else you would not have asked the question!).

[Guest [ou...]]

 

3. In a game of Russian roulette, which player should go 1st and why?

 

edited...

 

??

 

i was thinking it is like the monty hall problem but i realized we do not have the option of switching chambers after the first shot! intuitively my answer then is "it does not matter who goes first" but i know it is wrong (else you would not have asked the question!).

 

Don't be so sure; after all, the point of easy questions is to be tricky.

[Guest [Kp...]]

Let us call the area of an "equilateral triangle" of N dots T(N).

2 copies of such a triangle form an Nx(N+1) rectangle.

It follows then that T(N)=N*(N+1)/2.

Q.E.D.

 

if by N you mean each side of the eq. triangle is N dots then

 

2 copies of T(N) = 2 * T(N) = 2 * √3/4 *  N²  = √3/2 *  N²  = N *√3/2 N <> N * (N+1)

 

what am i missing?

[Guest [Kp...]]

2. Given an event A, find the minimum of 1/P(A)+1/P(A')

 

we know P(A) + P(A') = 1

 

let F = 1/P(A) + 1/P(A') = (P(A) + P(A')) / P(A)*P(A') = 1 /  P(A)*P(A')

 

we need to find the highest denominator (P(A)*P(A')) to get lowest F

 

let P(A) = x

then P(A') = (1-x)

we need to find maximum value f(x) = x(1-x)

the maximum value is 0.25 as per parabola graphed by google calc. 

 

thus F or the minimum is 1/0.25 = 4

 

(apologies for the inelegant proof)

 

edit. we can use brute derivation for slope 0 to calculate maxima f(x), or, prove x = 0.5 is by eliminating redundancy -- what we do when we code for primes between 1 and 100 -- we cross out multiples of 2 (stack reduced by 1/2), multiples of 3 so on...

 

[Guest [ba...]]

A few 'easy' ones...

 

1. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

 

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

 

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

[[ma...]]

1) bat=1.05; ball=0.05

 

2) 5 min, assuming each machine makes one widget

 

3) 47 days

[Guest [ba...]]

1) bat=1.05; ball=0.05

 

2) 5 min, assuming each machine makes one widget

 

3) 47 days

 

Congratulations - you just aced the Cognitive Reflection Test.  Only one in six gets all of these correct.  The math is easy but it's way tempting to make a gut decision instead of thinking it through.

[Guest [ou...]]

1. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

 

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

 

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

 

1. $0.05 (system)

2. 100/20=5min (100 machines make 100/5 widgets/min)

3. 47d (think backwards)

 

Edit: magnesi beat me to the answer.

[[ma...]]

Yeah Take that one, benzo brain fog

[Guest [ou...]]

Let us call the area of an "equilateral triangle" of N dots T(N).

2 copies of such a triangle form an Nx(N+1) rectangle.

It follows then that T(N)=N*(N+1)/2.

Q.E.D.

 

if by N you mean each side of the eq. triangle is N dots then

 

2 copies of T(N) = 2 * T(N) = 2 * √3/4 *  N²  = √3/2 *  N²  = N *√3/2 N <> N * (N+1)

 

what am i missing?

 

Any 2 sides share a dot, which is why I said "equilateral triangle",

which in turn is why 2 copies form a rectangle rather than a rhombus.

 

Take a look at this:

 

http://mathsisfun.com/algebra/images/triangular-number-3.gif

[Guest [ou...]]

2. Given an event A, find the minimum of 1/P(A)+1/P(A')

 

we know P(A) + P(A') = 1

 

let F = 1/P(A) + 1/P(A') = (P(A) + P(A')) / P(A)*P(A') = 1 /  P(A)*P(A')

 

we need to find the highest denominator (P(A)*P(A')) to get lowest F

 

let P(A) = x

then P(A') = (1-x)

we need to find maximum value f(x) = x(1-x)

the maximum value is 0.25 as per parabola graphed by google calc.

 

thus F or the minimum is 1/0.25 = 4

 

(apologies for the inelegant proof)

 

edit. we can use brute derivation for slope 0 to calculate maxima f(x), or, prove x = 0.5 is by eliminating redundancy -- what we do when we code for primes between 1 and 100 -- we cross out multiples of 2 (stack reduced by 1/2), multiples of 3 so on...

 

Correct again.

 

Min=4 when P(A)=1/2=50%

 

Elegant proof without calculus:

 

x(1-x) = x-x^2 = 1/4-1/4+x-x^2 = 1/4-(x-1/2)^2 <= 1/4

 

F = 1/x(1-x) >= 1/(1/4) = 4

[Guest [ou...]]

Problem: Which player should go 1st in a game of Russian roulette?

 

Assumptions & Rules:

1. There are 2 players.

2. The revolver contains 1 bullet.

3. The cylinder consists of 6 chambers, though you could try generalizing to N.

4. Each player takes a shot at the opponent in turn.

 

Variants:

1. The game ends after 2 turns, possibly resulting in a draw.

2. The game does not end until we have a winner and the cylinder is

either A. spun after each turn or B. not.

[Guest [Kp...]]

Let us call the area of an "equilateral triangle" of N dots T(N).

2 copies of such a triangle form an Nx(N+1) rectangle.

It follows then that T(N)=N*(N+1)/2.

Q.E.D.

 

if by N you mean each side of the eq. triangle is N dots then

 

2 copies of T(N) = 2 * T(N) = 2 * √3/4 *  N²  = √3/2 *  N²  = N *√3/2 N <> N * (N+1)

 

what am i missing?

 

Any 2 sides share a dot, which is why I said "equilateral triangle",

which in turn is why 2 copies form a rectangle rather than a rhombus.

 

Take a look at this:

 

http://mathsisfun.com/algebra/images/triangular-number-3.gif

 

you have assumed that each dot denotes a square of area 1 sq. 

 

if you make a similar arrangement using dots/circles to arrange them as an equilateral triangle or rhombus ((T(N) X 2) - area N dots/squares), then you have to subtract the area of the open spaces because dots or squares of 1 sq area will not tile into an equilateral triangle densely.

 

[Guest [Kp...]]

Elegant proof without calculus:

 

x(1-x) = x-x^2 = 1/4-1/4+x-x^2 = 1/4-(x-1/2)^2 <= 1/4

 

F = 1/x(1-x) >= 1/(1/4) = 4

 

neat! this is using (a - b)² = a² - 2ab + b²

 

now why don't these things come automatically to my mind?  (it means that i suck at pattern recognition... chess grand masters recall by pattern recognition, not by exact board configurations -- ref. a bbc documentary.)