Why doesn't order matter in this

  • Thread starter aaaa202
  • Start date
  • Tags
    Matter
In summary: This means that the order in which we choose the balls does not affect the probability of getting a particular combination. In other words, the order is independent. In summary, when choosing balls from a jar with black and white balls, the probability of getting a specific combination is the same regardless of the order in which the balls are chosen. This is because multiplication is commutative and the numerator and denominator of the probability calculation will always be the same. This concept is fundamental in combinatorics and holds true for any number of elements, not just black and white balls.
  • #1
aaaa202
1,169
2
Say you have two black balls and two white in a jar and you pick 3 at random. When one is picked it is taken out of the jar.
I want to know the probability of getting for instance WWB. That must be:

P(WWB) = 1/2*1/3*1 = 1/6

Now say you want WBW. Then you get:

P(WBW) = 1/2 * 2/3 * 1/2 = 1/6

Amazingly enough the probabilities are independent on the order in which you choose your balls. This result is kind of amazing for me and not really intuitive. So far it stands more or less as a defining property of this kind of combinatorics.
Is it intuitive for you that the order really is independent? What argument would you give for this?
And lastly, perhaps even more interestingly: Can anyone prove that if you general have NA blue elements and NB red elements then the number of ways of getting a blue and b red elements is independent of order. I don't know if it's an established theorem - if so, where can I read more about it?
 
Physics news on Phys.org
  • #2
Just look at those calculations: in the first you have the fractions 2/4, 1/3, and 2/2. In the second, the fractions are 2/4, 2/3, and 1/2. Do you see that, while the fractions are different, the three numerators and three denominators are the same?
Do you see why that is true? Do you see that, because the numerators are the same and the denominators are the same, the products are the same?
 
  • #3
I see it yes. Why it's true I'm not sure.
 
  • #4
Ivy, can you please help on the intuition for it?
 
  • #5
The order does not matter in general case.

Suppose you have [itex]b[/itex] black balls and [itex]w[/itex] white balls. The total amount of balls is [itex]n=b+w[/itex].
Consider a sequence {B,W,B,B,...,B,W}. The probability that this sequence occurs is:
[itex]\frac{b}{n}[/itex][itex]\frac{w}{n-1}[/itex][itex]\frac{b-1}{n-2}[/itex][itex]\frac{b-2}{n-3}[/itex]...[itex]\frac{b-(b-1)}{2}[/itex][itex]\frac{w-(w-1)}{1}[/itex]
Note that the numerator of this sequence is [itex]b(b-1)...(b-(b-2))(b-(b-1))w(w-1)...(w-(w-2))(w-(w-1))[/itex] and the denominator is [itex]n(n-1)...(n-(n-2))(n-(n-1))[/itex].
You should notice that whenever you want to choose [itex]b[/itex] black balls and [itex]w[/itex] white balls, the numerator and the denominator will be the same. The order of the elements of both products will be different, but you know that scalar multiplication is commutative (i.e. order does not matter).
 
  • #6
aaaa202 said:
Ivy, can you please help on the intuition for it?

The fundamental ideas are the we can multiply fractions by multiplying numerators and denominators separately and that multiplication is commutative. That is, we can rearrange the numerators and denominators as we please.
 

FAQ: Why doesn't order matter in this

1. Why is the order of operations important in math?

The order of operations is important in math because it provides a set of rules for solving mathematical expressions in a consistent and unambiguous manner. This helps to avoid confusion and ensures that all individuals arrive at the same answer when working on the same problem.

2. Why doesn't order matter in addition and multiplication?

In addition and multiplication, order does not matter because these operations are commutative. This means that the order in which the numbers are added or multiplied does not affect the result. For example, 2+3 and 3+2 both equal 5, and 2x3 and 3x2 both equal 6.

3. Why does order matter in subtraction and division?

In subtraction and division, order does matter because these operations are not commutative. This means that the order in which the numbers are subtracted or divided can affect the result. For example, 5-3 and 3-5 are not equal, and 15÷3 and 3÷15 are not equal.

4. Why is the order of elements important in chemical reactions?

The order of elements is important in chemical reactions because it determines the final product of the reaction. In a chemical equation, the reactants (starting materials) are written on the left side and the products (resulting materials) are written on the right side. The order in which the reactants are combined can affect the chemical reaction and the resulting product.

5. Why is the order of experiments important in the scientific method?

The order of experiments is important in the scientific method because it allows for a systematic and organized approach to testing a hypothesis. By following a specific order, scientists can eliminate potential variables and ensure that their results are accurate and reliable. This helps to establish a cause-and-effect relationship between the variables being studied.

Similar threads

B Relative And Absolute Probability (Probability Of Picking A Red Ball)
Replies
8
Views
467
MHB Calculate ways to form a committee of 3 from 8, DIRECTLY WITHOUT ÷?
Replies
1
Views
1K
MHB Partial Order Relation where the Set is not Necessarily Finite
Replies
6
Views
2K
Dragging colored balls off box in succession
Replies
9
Views
1K
I Question Regarding Linear Orders
Replies
28
Views
5K
I Probability of White Ball in Box of 120 Balls: Solved!
Replies
3
Views
2K
MHB Game Night : Number of combinations
Replies
14
Views
1K
I Statistical problem of drawing colored balls from boxes
Replies
4
Views
2K
I Finding the average time with given probability
Replies
3
Views
1K
B Conditional Probability, Independence, and Dependence
Replies
10
Views
2K

Hot Threads

Recent Insights

Back
Top