Question about Using the Formula for Combinations in Probability

In summary, the conversation discusses 12 refrigerators that have been returned due to a high pitched oscillating noise, with 5 of them having defective compressors and the other 7 having less serious problems. The question asks to compute the probability of having one defective compressor in a random sample of 6 fridges, as well as the expected value of X (the number of defective compressors among the first 6 examined) and the expected value of X^2. The relevant equation is (n choose k) = n!/k!(n-k)!.
  • #1
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Homework Statement

Each of 12 refrigerators of a certain type has been returned to a distributo because of the presence of a high pitched oscillating noise. Suppose that 5 of these 12 have defective compressors and the other 7 have less serious problems. If they are examined in random order, let X = the number among the first 6 examined that have a defective compressor. Compute the following:

P(x = 1)

E(X)

E(X^2)



Homework Equations

(n choose k) = n!/k!(n-k)!



The Attempt at a Solution

My random sample is 6, and there are 12 choose from. But I cant' seem to come up with a combination that makes any sense. (I'm just speaking specifically about trying to find P(x=1)
 
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  • #2
To get one bad compressor in a sample of 6, you have to choose 1 from the group of 5 that have bad compressors and 5 from the group of 7 that don't. Then to get the probability you divide by the number of ways to choose 6 from the 12 sample total. How do you express that in combinations language?
 

1. How is the formula for combinations used in probability?

The formula for combinations is used in probability to determine the number of possible outcomes when selecting a certain number of items from a larger set, without regard to order. This is useful in calculating the probability of events such as drawing a specific hand in a card game or selecting a certain number of colored marbles from a bag.

2. What is the difference between combinations and permutations in probability?

In probability, combinations and permutations both involve selecting items from a larger set. However, combinations do not take into account the order of the selected items, while permutations do. This means that combinations are used when order does not matter, and permutations are used when order does matter.

3. How is the formula for combinations derived?

The formula for combinations, nCr = n! / (r!(n-r)!), is derived from the fundamental principle of counting and the concept of factorial. The numerator represents the total number of ways to arrange a set of n items, and the denominator removes the duplicate arrangements when selecting r items from that set.

4. Can the formula for combinations be used for any type of problem in probability?

Yes, the formula for combinations can be used for any problem that involves selecting a certain number of items from a larger set, without regard to order. This includes problems in card games, probability experiments, and other situations where the order of selection does not affect the outcome.

5. What are some real-life applications of the formula for combinations in probability?

The formula for combinations has many real-life applications, including predicting the chances of winning in games of chance, analyzing outcomes in genetics and biology, and determining the likelihood of certain events in sports and finance. It is also commonly used in computer science for algorithms and data analysis.

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