Probability

1/120

there are three spots and six numbers on a dice

so 6 numbers on the first spot and 5 numbers on the second spot and 4 in the third spot just multiply them

If the four letters "A" are to be together, "AAAA", then it's like having four different

letters; AAAA, L, B, M.

The number of different arrangements (permutations) of the 7 letters in the word

"ALABAMA" putting the four As together are;

4! =

4x3x2x1 =

24

from itertools import permutations

from string import join

arrangements = [ ]

for p in permutations ( 'ugccsir', 6 ) :

arrangement = join ( p, '' )

if 'ui' in arrangement :

continue

if not arrangement in arrangements :

arrangements . append ( arrangement )

print len ( arrangements )

The above Python code gives a result of 2,220.

Here are two ways you can solve this, depending on if order matters or order does not matter:

Order of the dice does matter:

Assuming that you roll a 1, 1, 1, 1, 4 is not the same as rolling a 1, 4, 1, 1, 1

Think of 5 blanks first (one for each six sided dice).[ dice 1 ] [ dice 2 ] [ dice 3 ] [ dice 4 ] [ dice 5 ]

Each dice has 6 sides, so you can have 6 different possibilities per dice. Fill in each blank with how manypossibilities. Then multiply them.

(6) x (6) x (6) x (6) x (6) = 65 = 7776 possibilities.

A few possibilities looks like (I'm not listing all 7776possibilities):

(1, 1, 1, 1, 1), (2, 1, 1, 1, 1), (3, 1, 1, 1, 1), (4, 1, 1, 1, 1), (5, 1, 1, 1, 1), (6, 1, 1, 1, 1), where (1, 1, 1, 1, 1) represents all five dice landing on a 6.

The answer is 7776 possibilities.

Order of the dice does not Matter

Now if 1, 1, 1, 1, 4 is the same as rolling a 1, 4, 1, 1, 1 then the problem is more complex. You can use the following formula (it's a binomial):

C(n+5, 5), where n is the amount of dice

C(5+5,5) = C(10,5)

How to use this:

C(A,B) = A! / [ (A-B)! B! ], where A! is read A factorial

What is a factorial:

n! = n x (n-1) x (n-2) ... x ... (n-n+1)

Example:

5! = 5 x (5-1) x (5-2) x (5-3) x (5-4)

5! = 5 x 4 x 3 x 2 x 1

Plug in your numbers;

C(10,5) = 10! / [ (10-5)! 5! ]

C(10,5) = 10! / ( 5! 5! )

( 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 ) / ( 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 x 5 x 5 )

= ( 6 x 7 x 8 x 9 x 10 ) / ( 2 x 3 x 4 x 5 ) = 30240 / 120 = 252 possibilities

The equation I gave you works for six sided dice problems. I have verified this answer numerically.

So the answers are:

7776 possibilities if order matters.

252 possibilities if order does not matter

With a single die:

There are 6 sides on a dice.

There are 3 sides that are odd (1, 3, 5)

3/6 = 1/2 = 50%

Therefore, a single die has a 50% chance of landing on an odd number.

If you have two dice:

I'll explain this the longer way, however there are short (and simpler explanations) that basically use the fact that a single die is 50%, therefore two independent dice will also be 50%.

Number of dice = 2Sides of a dice = 6

Odd sides: 1, 3, 5, so Number of odd sides on a dice = 3

Even sides: 2, 4, 6, so Number of even sides = 3

Total combinations with two 6 sided dice = (sides on dice x sides on dice) = 6 x 6 = 36

We need to take the fact that:

An odd number + an odd number = even
An even number + an even number = even

An odd number + an even number = odd <--- we want this case

Odd combinations on a dice:

First dice is even, second dice is odd (even sides x odd sides):

(3) x (3) = 9

First dice is odd, second dice is even (odd sides x even sides):

(3) x (3) = 9

Total odd combinations:

9 + 9 = 18

Divide odd combinations by total combinations to get the probability:

18 / 36 = 1/2

You have a 50% chance to roll an odd number on a single roll of two six sided dice.

It means that Gerrymandering is less easy.
Compact: Think of dropping a big spoonful of honey on a table. The result will be a round blob. We would say that the area of contact between the honey and the table is compact. In contrast, if you were to drop a spoonful of water from a couple of inches above the table it would splash out. There would be 'arms' of water that would spread out from the central wet area on the table. The area of contact between the water and the table would not be considered compact.

Contiguous: The city of New York is not contiguous. There's the mainland and there's Long Island, and these pieces are separated by salt water. In contrast the state of Idaho is contiguous--because it's all one big area.

Equal in population: obvious.

Intuitively, a standard deviation is a change from the expected value.

For the question you asked, this means that the change in the "results" doesn't exist, which doesn't really happen. If the standard deviation is 0, then it's impossible to perform the test! This shows that it's impossible to compute the probability with the "null" standard deviation from this form:

z = (x - µ)/σ

If σ = 0, then the probability doesn't exist.

A large sample (n > 25) and p, the probability of success on each trial = around 0.5 (0.35 to 0.65).

Independence is already assumed for it to be binomial.

A large sample (n > 25) and p, the probability of success on each trial = around 0.5 (0.35 to 0.65).

Independence is already assumed for it to be binomial.

A large sample (n > 25) and p, the probability of success on each trial = around 0.5 (0.35 to 0.65).

Independence is already assumed for it to be binomial.

A large sample (n > 25) and p, the probability of success on each trial = around 0.5 (0.35 to 0.65).

Independence is already assumed for it to be binomial.