Probability involving combinations

In summary, the conversation discusses a probability question involving seating arrangements of two couples and one single person. The method given involves calculating the number of ways in which no couples sit together, which can be done by seating the single person in different places and considering the restrictions for the remaining seats. The questioner suggests finding an easier way to solve the problem, but the responder points out that the reverse question (finding the number of ways in which at least one couple sits together) may not be easier and suggests using combinations to solve the problem.
  • #1
Eagle784
6
0
Surely there's an easier way to do this question than the method given?

Q: Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

A: Yes. Let's call the first couple C and c, the second couple K and k, and the single person S. Let's seat S in different places and figure out the possible ways to have no couples sit together. If S sits in the first seat, any of the remaining four people could sit next to S. However, only two people could sit in the next seat: the two who don't form a couple with the person just seated). For example, if we have S K so far, C or c must sit in the third seat. Similarly, we have only one choice for the fourth seat: the remaining person who does not form a couple with the person in the third seat. Because we have seated four people already, there is only one choice for the fifth seat; the number of ways is 4 × 2 × 1 × 1 = 8. Because of symmetry, there are also 8 ways if S sits in the fifth seat. Now let's put S in the second seat. Any of the remaining four could sit in the first seat. It may appear that any of the remaining three could sit in the third seat, but we have to be careful not to leave a couple for seats four and five. For example, if we have C S c so far, K and k must sit together, which we don't want. So there are only two possibilities for the third seat. As above, there is only one choice each for the fourth and fifth seats. Therefore, the number of ways is 4 × 2 × 1 × 1 = 8. Because of symmetry, there are also 8 ways if S sits in the fourth seat. This brings us to S in the third seat. Any of the remaining four can sit in the first seat. Two people could sit in the second seat (again, the two who don't form a couple with the person in the first seat). Once we get to the fourth seat, there are no restrictions. We have two choices for the fourth seat and one choice remaining for the fifth seat. Therefore, the number of ways is 4 × 2 × 2 × 1 = 16. We have found a total of 8 + 8 + 8 + 8 + 16 = 48 ways to seat the five people with no couples together; there is an overall total of 5! = 120 ways to seat the five people, so the probability is = . Whew!

Thanks for your help.
 
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  • #2
Try the other way round, ie how many ways are there so that the two couples always sit as a couple.
 
  • #3
That's not much easier. First of all, the reverse would be, at least one couple sits as a couple. Second of all, isn't there a way to do this using combinations?
 

FAQ: Probability involving combinations

1. What is the difference between combinations and permutations?

Combinations refer to the number of ways to select a subset of items from a larger set, without considering the order of the items. Permutations, on the other hand, refer to the number of ways to arrange a set of items in a specific order.

2. How do I calculate the number of combinations?

The formula for calculating the number of combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items being selected. This formula is also known as the "choose" formula, as it represents the number of ways to choose r items from a set of n items.

3. How is probability involved in combinations?

Probability is used to calculate the likelihood of a specific combination occurring out of all the possible combinations. The probability of a combination is calculated by dividing the number of favorable outcomes (i.e. the specific combination) by the total number of possible outcomes.

4. Can you give an example of probability involving combinations?

Sure, let's say we have a bag of 10 marbles, 5 of which are red and 5 of which are blue. If we randomly select 3 marbles from the bag without replacement, what is the probability of getting 2 red marbles and 1 blue marble? The number of combinations of 2 red marbles and 1 blue marble is 5C2 * 5C1 = 10 * 5 = 50. The total number of combinations when selecting 3 marbles from 10 is 10C3 = 120. Therefore, the probability is 50/120 = 5/12 or approximately 0.42.

5. How is the concept of combinations applied in real life?

The concept of combinations is used in various fields such as statistics, finance, and computer science. In statistics, combinations are used to calculate the probability of certain events occurring, which can be used to make informed decisions. In finance, combinations are used in portfolio diversification and risk management. In computer science, combinations are used in algorithms for data mining and machine learning.

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