Toss a coin and see...
The chances of a tossed coin falling head or tails is almost 50-50 with an extremely small chance of it falling on its edge. Therefore what should be the dimensions of a coin that has a probability of 1/3 to fall heads, 1/3 to fall tails, and 1/3 on its edge?
What is one supposed to do after reading this, You could of course pretend you haven’t read it but then who would know you didn’t?
What is one supposed to do after reading this, You could of course pretend you haven’t read it but then who would know you didn’t?
Labels: friday special, logic, thinktank
posted by Rajesh Lal at
9:23 AM
32 Comments:
The only way that I can see that happening is if the coin was in the shape of a cube which has the same dimensions on each side.
This is a diagram of each side of the cube:
T=Tails
H=Heads
S=Side
[] [] [] [] [] []
T H S T H S
Seeing this, it is safe to say that you would get 2/6 chance of landing on one side, and reduced would give you what?
1/3 chance to land on each side?
Did I mess anything up?
I'm thinking that it's not really a cube, but a cylinder, w/ the length being at least the same measurement as each face.
or maybe it might be impossible. i picture that cylinder flipping in the air, and i realize that it actually has twice the chance of landing on its side than on either of the faces. heads, side, tails, side, head, side, tails, side...so on, so forth. You'd need a shape that has 3 faces, and no such shape exists, right? not even a pyramid. that's my two cents.
actually, a cube would make make the probability to land on any side 1 in four...the answer would be a pyramid...
I do not believe that it could be a pyramid because you would be missing either a head or tail and you would have four sides giving you a probability of 1/5. It also could not be a cube because there are six sides therefore it has a probability of 1/6.
The answer would have to be a cylinder. There is there surfaces in a cylinder therefore the probability is 1/3.
correcting myself, a pyramid has four sides and a base...5 surfaces total. That is where the probability of 1/5 came from.
All working determined as if Pi were 3.14 exactly
If the coin has a radius of 5cm (from the centre to the outer rim) then using the formula Area= Pi x radius squared, the area of this face is 78.5cm squared.
Now as this face has a radius of 5cm, its diameter is 10cm. Diameter x Pi = 31.4cm - this is the length of the outer rim of the coin.
78.5 / 31.4 = 2.5
78.5cm, or the total area of each coin face, divided by 31.4, the coins diameter, = 2.5cm. This means that the distance between the 2 faces of the coin must be 2.5cm in order to give each face an equal 1-in-3 chance being landed on.
The simple(r) answer is that if the coins diameter is x, then the distance between the 2 faces of the coin must be x over 2 - or half of x
Ps. Could someone please confirm or disprove my answer - thanks =]
I also believe that it's a Cylinder or a Thick Coin.
In order to have probability of 1/3, the coin thickness should be 1/2 the radius of the coin.
Therefore, the coin consist of Face, Tail & Side that have the same Area (=phi*r*r = 2*phi*r*t).
sorry, i did mean half the radius, not the diameter
I disagree with Daniel. I do not believe it could be a cylinder.I do agree that the odds for each side would be a lot closer, but a spinning cylinder/coin would have to be flipped in a pattern. the traditional way of flipping a coin will increase the probability of it landing on a face side (heads or tails.) By flipped from the round side, you will increase the probability of it landing on that side.
The only shape I can picture that would work, would be a triangular "dreidel" with both ends being cone shaped and shorter in length than the 3 sided body. giving you 3 equal sides all equaly probable when flipped or spun.
I could be wrong, but im young. Ill get over it.
can someone please clarify why it is not a cube. a cube has 6 sides. 2 for heads, 2 for tails, and 2 for side. prob = 1/3.
please clarify why this is wrong
well the question asks for odds 1/3, and i know that 2/6 is the same and the cube is a possibility - but its does say "and 1/3 to fall on its EGDE", not multiple sides labelled "edge". so i think its sort of asking for a 3 sided shape, my opinion could easily be wrong
It's not a Cube because the question is "what should be the dimensions of a COIN that has a probability of 1/3 to fall heads, 1/3 to fall tails, and 1/3 on its edge?"
a Cube is NOT a Coin.
I think that blcros97 is totally right. That cube/coin would sure be annoying in your pocket or wallet tho!
the edge of the coin would have to have the same area as each of the sides of the coin individually.
as was mentioned earlier, it does say what are the DIMENSIONS of a COIN that has... so the question is not what SHAPE, its what DIMENSIONS so again, the A of E = A of H = A of T
area of edge is area of heads is area of tails
If the height of the coin is equal to the radius, it will have an equal chance of landing on its side or one of its faces.
Another possible shape would be something like a Toblerone bar, although there is a slight chance of it landing on one of its ends, but you could always round off the ends.
i believe that it would be a hollow square, each wall to be any size that you want ..as long as they are all the same size.. i will attempt to draw a pic using the first letter of the side name as the actual building blocks.. so please dont make to much fun of my art work..lmao(I KNOW THAT THEY ARE NOT TO SCALE AND THE ACTUAL NUMBER OF LETTERS ARE NOT THE CORRECT NUMBER OF LETTERS .. I JUST DID IT LIKE THIS TO GIVE YOU A MENTAL IMAGE..
H T
H T
SSSSSSS
ok if you think about the question, it says heads tails or SIDE, well on a coin there is only 1 side, and it has no start and no finish... so i believe that the shape would be a U.. please let me know what you think..try not to be to hard on me..lmao
ok my pic didnt come out as i planned.. let me try to do it this way.. and just imagine that the (.'s) are not there
H....T
H....T
H....T
SSSSSS
HWDIV has a good point that it may not be a cylinder, but if you make a three sided dreidel, it doesn't really have an 'edge' for it to land on since the edge looks just like another side.
Lets Assume its a cylinder and that you only 'flip' the coin the way you normally flip it, with thumb and forefinger flipping it.
Obviously there is a thickenss of the cylinder to solve this because if you flip a pencil which is a very thick cylinder, it will almost always land 'on edge'.
So assume the coin lands at an angle alpha, and then falls over based on where its center of gravity is relative to the pivot point. See the diagram below.
..........-\
......\-....\
.......\..o..\
.........\....-\
..alpha..\-
--------------------------
You want the center to line up right over the pivot point when alpha =66.6 degrees. That way, ignoring bouncing, when it first touches down leaning to the left side it has a 2/3 chance of landing on one face and 1/3 chance of landing on edge. Similarly if it first touches down leaning to the right, it will have a 2/3 chance on landing on the other face and 1/3 chance of landing on the edge.
Drawing a little traingle where D is the diameter of the coin and L is the length (Depth) you get
......L -|
.....\-..|
......\..|
.....D.\.|
.........\|
...66.6..\
So, Tan(33.33) = L/D.
Depth of coin is .658 times the diameter.
I just rounded the side to .5 times the diameter of the coin
It could be a shape with three surfaces that are NOT PLANES.
The physical dimensions are irrelevant here, so it may be not the right answer.
its a one in a million chance of a coin landing on its edge i learned that from the twilight zone episode a penny for a thought lol
it would have to be in a shape of a cilinder
heads weighs more than tails, so it has a bettter chance to land on heads!
It most certainly could be a cube, since the word COIN is just that. A word that we have designated to mean something representing a monetary worth. So if we designated a monetary worth to a cube (I'm sure this has existed in some culture in history) then the cube can be called a coin.
I could also be full of crap too though. (haha)
I'd say either a non-existing shape in a 3d world or a cube, with two faces with heads, two faces with tails, and finally two faces with "edge".
Is this a weight distribution problem?
Assuming the coin is a cylindrical shape which has been tossed inside a vacuum, the main force acting on it will be gravity. Therefore, its mass must be distributed equally through the vertical and horizontal axes in order for the coin to have an equal probability of landing on one of its faces or its side.
My way of approaching this would be to calculate the area of the plane in each axis, as gravity would act on a plane rather than surface area. The weight distribution acting through each plane must be the same.
Let r be radius and x the thickness of the coin:
(massxgravity)/(pi)r2 = (massxgravity)/2rx
Therefore (pi)r2=2rx, so x=(pi)r/2
This implies that the thickness of the coin is approximately 1.6 times its radius which, intuitively, must be somewhere near the mark? I wait to be shot down in flames!
ANS: if the coin is flipped to a horizontal plane, ignoring air friction and bouncing,the ratio= 1.73
Otherwise if any orientation is possible , then ratio = 2.83.
We did the problem recently, it involves angles mainly ...u can figure out how to get the answer :)
Sorry i forgot to say that the 'ratio'(1.73) is the ratio of the diameter to the thickness
assuming he dia to be 1", one side is 22/7 sq inches and so is the other side, therefore the thickness of the round coin must be= 22/7 x 7/44 time the radius. This comes .5 times radius, in this case .5 sq inches
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