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# Don't think too quickly

SealPoet

1
Thu 27 Mar, 2003 05:25 am
Okay you bit twiddler...

I have a random number between one and three, inclusive. What's the chance that the number is a three?

Same question reduced.

Flip two binary coins, the possible results are
00
01
10
11
.... or convert to decimal 0, 1, 2, 3

We know that there is a head, so 00 (or 0) is out.
0 Replies

Craven de Kere

1
Thu 27 Mar, 2003 11:01 am
That one gives more leeway to "creative interpretation".

If you are asking "what is the probability with the following rules" it's 1/3:

a) If 00, flip again.

AND

b) Both coins are flipped before the question is asked and both flips are repeated if the result is 00.

If you do it this way it is 1/2:

Flip one until you get heads.

Then flip the other.

edit: added quotation marks to correct grammar.
0 Replies

Monger

1
Thu 27 Mar, 2003 12:23 pm
Now that that one is cleared up, here's another "don't think too quickly" type of riddle...

Compare the numbers 0.99999... (infinitely many 9s) and 1. Which of the following statements is true? Why?

0.99999... < 1
0.99999... = 1
0.99999... > 1
0 Replies

BillW

1
Thu 27 Mar, 2003 12:38 pm
<, it'll never quite get there
0 Replies

Monger

1
Thu 27 Mar, 2003 01:01 pm
Bill, there's a method to remove recurrence from a decimal:

Code:```10 x .9999... = 9.9999... 9.9999... - .9999... ----------- 9 9/(10-1) = 1 --------------- 0.99999... = 1 ---------------```

2 other simple examples:

1/3 = .333333......
2/3 = .666666......
3/3 = .999999......

And,
1 - 0.9999... = 0.0000... (If you take the difference between two numbers, and the difference is 0, then they have to be the same number.)
0 Replies

BillW

1
Thu 27 Mar, 2003 01:06 pm
Ah, but that's rounding - still didn't get there, but we'll just say it did! What's the largest number?
0 Replies

Monger

1
Thu 27 Mar, 2003 01:20 pm
None of the explanations I gave use rounding. Remember the ... means it's continued to infinity. The first example, to me, is the most clear. The second example shows 1/3 = .333....... exactly. The .... is an indication that the 3's continue on to infinity. This is not an aproximation, it is the exact value, represented by the dots to save time from writing an infinite number of 3's. But it is the exact answer.
0 Replies

BillW

1
Thu 27 Mar, 2003 01:25 pm
And 3 x .333... = .999..... which is not = 1, it approaches 1.

What is the largest number?
0 Replies

cicerone imposter

1
Thu 27 Mar, 2003 01:32 pm
Monger, In bookkeeping, it's equal to 1.0.
0 Replies

BillW

1
Thu 27 Mar, 2003 01:38 pm
c.i., that's because we record at 2 decimals and round at 3!
0 Replies

Monger

1
Thu 27 Mar, 2003 01:53 pm
BillW wrote:
And 3 x .333... = .999..... which is not = 1, it approaches 1.
What is the largest number?

There is no largest number.
If the sum to infinity of 3/10 + 3/100 + 3/1000 + ... is not equal to 1/3, what is it equal to? If it is EXACTLY equal to 1/3, then 3/3 = 0.999...

Don't confuse "doing something many, many times" and "doing something an infinite number of times." Yes you get closer every step you do, and as you are approaching infinity you are approaching 1, that is true. However, when/if you hit infinity you hit 1 also.

Math works on paper. Sometimes you have to believe in what your calculations prove and show, otherwise we have to imagine that math works completely differently than it does.

Again, take a look at the first example I gave:
Code:```10 x .9999... = 9.9999... 9.9999... - .9999... ----------- 9 9/(10-1) = 1 0.99999... = 1```
Note that no rounding or approximation is used there at all.

Here's a good link I found on this topic. http://mathforum.org/dr.math/faq/faq.0.9999.html
0 Replies

Monger

1
Thu 27 Mar, 2003 02:34 pm
Here's an interesting excerpt from one of the pages at that web site:

Quote:
There's no doubt that this equality is one of the weirder things in mathematics, and it is intuitive to think: No matter how many 9's you add, you'll never get all the way to 1.

But that's how it seems if you think about moving toward 1. What if you think about moving away from 1?

That is, if you start at 1, and try to move away from 1 and toward 0.99999..., how far do you have to go to get to 0.99999... ? Any step you try to take will be too far, so you can't move at all - which means that to move from 1 to 0.99999..., you have to stay at 1.

Which means they must be the same thing!
0 Replies

TechnoGuyRob

1
Wed 2 Apr, 2003 04:06 pm
It's =
Think this way:

1/1 = 1
2/2 = 2
3/3 = 3
.
.
.
999..../999.... = 1

1/3 = 0.33...

2/3 = 0.66...

3/3 = 0.99...

therefore, with calculus it can be proven that the limit as n approaches infinity of n/n = 1, which means 3/3 = 1 for certain, and therefore 3*1/3 = 3/3 = 1!
0 Replies

jespah

1
Wed 2 Apr, 2003 04:19 pm
Welcome, TechnoGuyRob! :-D
0 Replies

cicerone imposter

1
Wed 2 Apr, 2003 05:24 pm
TechNoGuy, WELCOME to A2K. Now that I understand a little math, and 3/3 = 1 or .99, because 2/3 = .66, is it still accurate to say 2/3 = .67? c.i.
0 Replies

TechnoGuyRob

1
Fri 4 Apr, 2003 10:03 am
N/A
1 is not equal to 0.99, nor is it equal to 0.9999999999999999. 1 is only equal to 0.999..., and 2/3 is only equal to 0.666...
Saying that 2/3 = 0.66 or 0.67, is only an approximation. But for example, for anyone who knows calculus, you can see this is a geometric serie:

0.9 + 0.09 + 0.009 + ....

A geometric series is a series that is defined as:

a + a*r + a*r^2 + a*r^3 + ....

In this case, a = 0.9, and r = 0.1, and according to definition, a geometric series approaches

a/(1-r)

when it's sum is added (assuming r < 1). Therefore, in this case

.9/(1 - 0.1) = 0.9/0.9 = 1
0 Replies

Blaine The Mono

1
Fri 4 Apr, 2003 12:07 pm
I really have to give it to Enthusiast !!!!!! Bravo!!!
1/3. You might initially think that the answer is 1/2, but not so. For two coins, there are four possible outcomes: HH, HT, TH, TT, since we know that at least one was a head, we can eliminate TT. Of the remaining three possibilities, only 1 allows the second head: HH. As mentioned by Enthusiast.
0 Replies

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