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#1 2006-02-27 16:50:12

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Probability and Statistics

PS # 1

Ten fair coins are tossed simultaneously. Find the probability of getting atleast seven heads.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2006-02-28 04:15:56

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Probability and Statistics

PS # 1

This is a binomial distribution, with 10 trials and a probability of success of 0.5.

The probability of at least 7 heads is P(7) + P(8) + P(9) + P(10).

This is worked out by 0.5*10 (10C7 + 10C8 + 10C9 + 10C10) = 11/64.


Why did the vector cross the road?
It wanted to be normal.

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#3 2006-02-28 04:41:45

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

Very good, mathsyperson. You're correct. up


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2006-02-28 20:08:07

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

PS # 2

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#5 2006-03-01 03:47:18

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Probability and Statistics

Ooh, interesting. Let's see... a chessboard has 4 corner squares, 24 edge squares and 36 centre squares.

On the corner squares, the chance of the other square being next to it is 2/63.
On the edge squares, the chance is 3/63 and on the centres, the chance is 4/63.

Combine this with the chances of the first square being each of the types and we get (4/64*2/63) + (24/64*3/63) + (36/64*4/63) = 224/4032 = 1/18.


Why did the vector cross the road?
It wanted to be normal.

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#6 2006-03-01 03:58:37

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

Excellent! Well done, mathsyperson! smile


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#7 2006-03-01 16:19:11

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

PS # 3

The probabitlity that a student will pass in Statistics examination is 2/3 and the probability that he will not pass in mathematics is 5/9. The probability that he will pass in atleast one of the examinations is 4/5. Find the probability of his passing in both the examinations.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#8 2006-03-02 04:31:26

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Probability and Statistics

P(AnB) = P(A) + P(B) - P(AuB).

Therefore, P(Passing both) = 2/3 + 5/9 - 4/5 = 19/45.


Why did the vector cross the road?
It wanted to be normal.

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#9 2006-03-02 16:43:36

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

ganesh wrote:

and the probability that he will not pass in mathematics is 5/9.

mathsyperson, did you notice that?
The probability that he will pass in mathematics is, therefore, 4/9.
Hence, the probability of passing both the subjects would be
2/3 + 4/9 - 4/5 = 14/45.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#10 2006-03-08 16:17:05

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

PS # 4

The probability of rain was 40%. If it rained, the Redskins had a 30% chance of winning, if it did not rain, they had a 55% chance of winning. Given that the Redskins won, what is the probability that it rained?


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#11 2006-03-09 05:01:25

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Probability and Statistics

Oh, silly me. I hate it when I make stupid mistakes from not reading the question properly. Ah well.

The probability of it raining and of them winning is 0.4*0.3 = 0.12.
The probability of it not raining and of them winning is 0.6*0.55 = 0.33.

Therefore, the probability of rain if they won is 0.12/(0.12+0.33) = 0.26666... = 27% (nearest %)


Why did the vector cross the road?
It wanted to be normal.

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#12 2006-03-09 16:13:39

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 46,753

Re: Probability and Statistics

Excellent, mathsyperson!


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#13 2009-10-11 09:35:44

maudish
Member
Registered: 2009-10-11
Posts: 1

Re: Probability and Statistics

Hi, could you please answer this problem?
In how many ways you can choose 2 white squares on a chessboard such that they are either in the same row or same column?

Thanks in advance.

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#14 2009-10-11 10:46:43

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Probability and Statistics

Hi maudish;

I believe the sequence looks like this:

For various n x n boards:
2 x 2 =0,
3 x 3 =4
4 x 4 =8,
5 x 5 =22
6 x 6 =36
7 x 7 = ?,
8 x 8 = 96

All by actual count. If i find a formula for n x n will post it.

Last edited by bobbym (2009-10-11 10:50:38)


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2009-10-12 14:59:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Probability and Statistics

Hi maudish;

Using the ordinary generating function for the rook polynomials on a r x c board.

The series can be eliminated because we only seek the coefficient of x^2 ( 2 rooks). Also since the board is square r = c. So.

The above is the number of 2 non attacking rooks on a c x c chessboard.

From a combinatorical argument and playing much spot the pattern.

We can solve for n and clean up:

Where n is the number of ways 2 rooks can be positioned on the white squares of a c x c chessboard when c is even. The above will generate the table given in the previous post, i.e.

c = 2 then n = 0
c = 4 then n = 8
c = 6 then n = 36
c = 8 then n = 96
c = 10 then n = 200
.
.
.

This is not a proof, just a conjecture. I have tested it for c = 16 by direct count. I suppose it might be proven by induction but the correct method is by partitioning the chessboard with it's forbidden black squares into disjoint boards and then using the rook polynomials to prove it. When I do that I will post it.

Last edited by bobbym (2009-10-12 15:17:55)


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#16 2011-02-28 17:42:16

G-man
Member
Registered: 2011-02-28
Posts: 16

Re: Probability and Statistics

If I'm not wrong.
Answer to question in first post.

Last edited by G-man (2011-02-28 18:22:56)


Maths!......sleep

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#17 2011-02-28 18:00:15

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Probability and Statistics

Hi G-man;

Welcome to the forum. Which question are you answering?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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