There are four married couples and eight seats. When they sit, what is the probability that husband sits adjacent to his wife?

The answer to this problem is 12/35

I can arrive at this answer when I try to calculate the probability of the complement and then subtract it from 1. i.e. 1 - Probability that one couple sit together - Probability that two couples sit together - Probability that three couples sit together - Probability that four coupls sit together

However when I try to calculate the probability straight, I can't do it right.

This is how I'm doing it:

Denominator will be 8! Building the numerator:

Numerator will be a product. The first seat can be occupied by 8 people, the second seat can be occupied by 7 people, but one of them could be the first seated person's spouse, so second term will be 6. Numerator so far: 8 x 6

Third seat can be occupied by five people but one of the could be the spouse of the person on the second seat, so third term will be 5. Numerator so far: 8 x 6 x 5

Similarly we get: 8 x 6 x 5 x 4 x 3 x 3 x 2

(Note that on the seven seat we get the number three again because we are sure that this person's spouse has been seated in one of the first four seats)

But by these calculations I get the answer 3/7

What am I doing wrong?

  • 4
    $\begingroup$ Shouldn't they tell you how the seats are arranged? In a circle so everyone sits with someone on their right and left? Are the seats even adjacent? $\endgroup$ – papercuts Feb 9 '13 at 10:03
  • 2
    $\begingroup$ Question unclear. Probability exactly one husband sits next to his wife? at least one husband? every husband? $\endgroup$ – Gerry Myerson Feb 9 '13 at 11:33
  • 1
    $\begingroup$ Note that if the 3rd person seated is the spouse of the 1st, then there are $5$, not $4$, choices for the 4th person. $\endgroup$ – Gerry Myerson Feb 9 '13 at 11:35

First, judging from your work, you misstated the question: it appears that you meant to ask for the probability that no husband sits next to his wife. It also appears that the seats are arranged in a row, not a circle. I will make these assumptions.

Your first three factors, $8\cdot6\cdot5$, in the numerator are fine. After that, though, you have to split the calculation into cases. Suppose that the first three people, in order, are A, B, and C. If A and C are a couple, then any of the remaining $5$ people can sit in the fourth seat. Your calculation, with a factor of $4$, is correct only if C’s spouse has not already been seated, i.e., C and A are not a couple. And the cases just proliferate after that, which is why you’re better off working with the complement.


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