It really is a great book - I don’t look the same way at zero anymore.
Probability.
If I flip a coin 1000 times I will get approx 500 heads and 500 tails. I seem to recall actually physically trying this some years ago and it did indeed work out.
But why?
Before each flip there is an equal chance that it will be a head as there is that it will be a tail. So how does the coin know that it has to even itself out over the course of the thousand flips?
If my first 400 flips came up heads then there is a good chance that my next 600 flips will turn up 100 heads and 500 tails.
How does the damn coin know?
Jojo, the damn coin doesn’t know. It is true that flipping a fair coin 1000 times you will get around a 500/500 split. However, you’re right, the coin has no memory. You can get results far off the 500/500 split but they’re highly unlikely.
If your first 400 flips come up heads, this is a highly unlikely event in and of itself. Probability is, however, that the next 600 flips will come up 300/300, so the end result is 700/300. Highly unusual, yes, but by assuming the first 400 are heads, you are in effect saying ‘if something highly unusual happens (400/0) then something highly unusual happens (700/300).’
Flipping a coin and cointing heads/tails constructs a ‘binomial distribution’. If you flip a coin 6 times you can write out all the possible results:
TTTTTT
TTTTTH
TTTTHT
TTTTHH
TTTHTT…
Even looking at the first results you can see that you only get 6 tails once of all the combinations (64), but you get 5 tails and one head 6 times. This is why you can say it is 6*more likely to get the result 1/5 than 0/6. As you get closer to 3/3 the odds (combination of flips that give you the result) increase.
It’s not. The 50/50 chance is fixed forever. The number of heads tosses does not at all change the heads/tails probability, it still remains even.
Just to clarify. Every time you toss the coin there will be a .5 chance it will be heads, this is independent of any other tosses.
Using Jojo’s If my first 400 flips came up heads, the chance the next flip will be heads is still .5 Therefore the most likely is that the of the next 600 flips, 300 will be heads and 300 will be tails.
There is no memory.
Bear in mind this is in a theoretical universe containing only a coin.
In real life physical stuff will affect the outcome (how you throw the coin etc.)
Sorry I should have previewed. Corrections
Using Jojo’s If: Using Jojo’s example If
Therefore the most likely is that the of the next: Therefore the most likely outcome is that of the next
Sorry.
includes a whole lotta things, because I’m just not very mathematically inclined.
Most recently, though, I have realized again that I don’t really get simple odds.
7-2 odds
5-4 odds
1-1 odds
I started to put forth what I THINK some of them indicate, but I won’t. Because I’m probably wrong.
If anyone chooses to answer this, could you explain it to me as though I were about 7 years old?
Thanks!
Stoid
Math Dolt
Going on memory here…
Suppose the odds of something are x-y odds. This means that there’s an y/(x+y) chance that it will occur, and a x/(x+y) chance that it won’t.
So an event with 7-2 odds has a 2/9 chance of occuring; an event with 5-4 odds has a 4/9 chance of occuring; and an event with 1-1 odds has a 1/2 chance of occuring.
When mathematicians couldn’t get the square route of a negative number, what did they do?: Invent a new type of numbers: imaginary.
Why don’t they do the same thing for when you divide numbers by 0? If they did that, it would make life much easier afterall, all numbers would equal eachother, light speed travel would be possible (mathematically speaking at least), hyperboles could finally intersect the X and Y axises…, the list goes on.
Why would it make things easier if all numbers equalled each other? That would actually be a disaster, as it can be shown that 0 != 1. If you can also show that 0 = 1, then you can show anything at all, and that’s no use.
Imaginary numbers weren’t just pulled out of the air, either. They arise in the solution of arbitrary cubic equations. In fact, it was the usefulness of the cubic formula that made people take negative and imaginary numbers seriously.
You may be interested in non-standard reals, which allow infinitely small quantities and their reciprocals. Be forewarned, however, that such things are not simple, to say the least.
Well, ultrafilter, I was being sarcastic about the “easier” part. But your point is taken.