How can permutations help with counting using probability?

In summary, the conversation discusses two approaches used in counting problems - combinations and probability. The example of a carton containing 12 toasters, 3 of which are defective, is used to demonstrate how these approaches are applied. The combinations approach is used to find the probability of exactly one defective toaster in four randomly sold toasters, while the probability approach uses permutations to calculate the total number of possible outcomes. The concept of hypergeometric probability is also briefly mentioned. The conversation ends with the clarification that the probability of a certain permutation does not account for all possible permutations with one defective toaster, and a simpler example is used to further illustrate this point.
  • #1
musicgold
304
19
Hi All,

I am trying to reconcile two approaches used in counting problems. The first approach uses combinations and the other uses probability. I understand the combinations approach, but not able to comprehend the probability approach. Consider the following example,

A carton contains 12 toasters, 3 of which are defective (D). If four toasters are sold at random, find the probability that exactly one will be defective.


By the combinations approach: 3C1 * 9C3 / 12C4 = 252/495 = 0.509

Now the probability approach :

I am purposely want to use here permutations and not combinations.

total possible permutations = 12P4 =11880

Another way of finding the total permutations is to find the chance of getting any permutation = 1/12 * 1/11* 1/10*1/9 = 0.000084. The reciprocal of this number gives the number 11880.

My confusion starts now. Consider the case DGGG, where G = Good, and D= Defective.

P (DGGG) = 3/12 * 9/11 * 8/10 * 7/9 = 0.127

If we divide this number by the probability of each permutation, 0.000084, we get 1512.
Thus, DGGG accounts for 1512 permutations or 1512 / 24 = 63 combinations.

I am having a hard time understanding how these 1512 permutations or 63 combinations look like.

Could you please help me with this?

Thanks,

MG.
 
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  • #2
What they look like? Imagine each defective toaster labeled [itex]\(D_1[/itex], [itex]D_2[/itex], or [itex]D_3[/itex] and each good toaster labled [itex]G_1[/itex], [itex]G_2[/itex], ..., [itex]G_9[/itex]. Now look at all possible ways of writing one D and three Gs: [itex]D_1G_1G_2G_3[/itex] is one possibility, [itex]G_4D_2G_7G_5[/itex] is another,[itex]G_1D_1G_3G_2[/itex] yet another. I'm not about to write all 1512 of those! Notice that [itex]D_1G_1G_2G_3[/itex] and [itex]G_1D_1G_3G_2[/itex] have exactly the same "toasters" just in a different order. Those would be counted as different "permutations" but are exactly the same toasters. If we count only those that have at least one toaster different, we get the 63 "combinations".
 
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  • #3
This is a case of hypergeometric probability. Look that up.
 
  • #4
musicgold said:
I am having a hard time understanding how these 1512 permutations or 63 combinations look like.

What you have implicitly done is the equivalent of painting a number on each toaster. Rather than having twelve toasters, three of them defective, you have twelve distinct toasters, labeled {D1,D2,D3,G1,G2,G3,G4,G5,G6,G7,G8,G9}. Suppose you draw four toasters at random. For example, suppose you draw, in order, G2,G9,G1,D2. Drawing G1, D2, G9, G2 in order is a distinct permutation but the same combination as the previous example.

The 1512 permutations you discussed are the permutations of four toasters chosen from the twelve that have a defective toaster as the first item and good toasters as the remaining three. One way to look at that is to split the set of four into the defective toaster and three good toasters. The defective toaster is easy: three permutations. The good toasters are easy, too: the number of permutations of three items chosen from a set of nine is 504. 3*504=1512.

Regarding the 63 combinations, that is not a valid thing to do. You are explicitly talking about an ordering. You have to look at the entire picture if you want to discuss combinations.
 
  • #5
Thank you everybody.

I think DH hit the nail on the head. What I was trying to understand was whether P(DGGG) accounts for a permutation such as GGDG ( with the third toaster being defective). And now I am clear that it does not account for that permutation. P(DGGG) accounts only for the permuations with the first toaster being defective and the next three being good. To account for all the permuations with one defective toaster, we need to consider P(GDGG), P(GGDG) and P(GGGD).

As I was struggling with this problem, I simplified it further to only choose 3 toasters from 6 total toasters ( 4 good, 2 defective). In that case,

P( any permutation ) =1/6*1/5*1/4 =0.0083

P(DDG) = 2/6 * 1/5 * 4/4 = 6.67%, and P(DDG) accounts for 8 permutations ( 6.67% / 0.0083 = 8).

These 8 permutations have the defective toasters in the first two places. I then realized that P(DDG) accounts for the following 8 permutations.

D1 D2 G1
D1 D2 G2
D1 D2 G3
D1 D2 G4
D2 D1 G1
D2 D1 G2
D2 D1 G3
D2 D1 G4

Please let me know whether my logic is correct. Thanks once again,

MG.
 

Related to How can permutations help with counting using probability?

1. How is probability used in counting?

Probability is used in counting to determine the likelihood or chance of an event occurring. It allows us to quantify uncertainty and make predictions about the outcome of a particular situation.

2. What is the difference between counting and counting using probability?

Counting involves simply determining the number of objects or events in a set, while counting using probability takes into account the likelihood of each object or event occurring and assigns a numerical value to that likelihood.

3. Can probability be used to count large numbers?

Yes, probability can be used to count large numbers by breaking down the counting process into smaller, more manageable parts and using probability principles to determine the likelihood of each part occurring.

4. How does the concept of independence relate to counting using probability?

The concept of independence is important in counting using probability because it allows us to assume that the outcome of one event does not affect the outcome of another event. This simplifies the counting process and allows us to use basic probability principles.

5. What are some real-world applications of counting using probability?

Counting using probability has many real-world applications, such as in finance and economics to predict stock market trends, in medicine to assess the risk of certain diseases, and in sports to analyze the performance of athletes. It is also used in weather forecasting, risk assessment, and quality control.

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