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Consider a group of 30 people: $20$ men and $10$ women.

In a $4$-seat table, what's the probability of choosing a man in the 3rd seat?

I thought of 30 1st seat, 29 for the 2nd seat, and because we may have 20-19-18 men to choose from for the 3rd seat depending on what happened before, I've decided to use conditional probabilities.

So, we get $P(3rd M|W,W)P(W,W)+...+P(3rd M | M,M)P(M,M)=2/3$.

However, I've seen a resolution where someone just did $P^{20}_1\times P^{29}_3/P^{30}_4=2/3$. Why is it possible to reason in this simplified way? I would think this way only if we were computing the probability for picking a man in the 1st seat.

Any help would be appreciated.

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2 Answers 2

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The number of seats at the table is irrelevant.

The fact that it is the third seat is irrelevant.

The probability is ${20 \over 30} = {2 \over 3}$.

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  • $\begingroup$ Thanks for your answer. But why is the number of the seat(3rd) irrelevant David? $\endgroup$ Commented Sep 11, 2017 at 22:10
  • $\begingroup$ You may relabel the seats in any way you want, and the problem doesn't change. You may even name them with non-numeric values, like President, VP, Treasurer, and Secretary. $\endgroup$
    – Momo
    Commented Sep 11, 2017 at 22:17
  • $\begingroup$ Are you saying that the 3rd seat may not correspond necessarily to the 3rd seat being filled. My doubt is why the filling order does not matter, irrespective of the number/name of the seat? $\endgroup$ Commented Sep 11, 2017 at 23:15
  • $\begingroup$ If the filling order had to be sequential, then the only way to solve this would be using the conditional probabilities? $\endgroup$ Commented Sep 12, 2017 at 0:08
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To answer strictly your question:

$P_1^{20}$ means that you choose a men out of the 20 to fill the third seat

$P_3^{29}$ means that you choose three people out of 29 men and women left after the first choice to fill the seats 1,2, and 4

$P_4^{30}$ means that you choose four people out of 30 men and women to fill the seats 1,2,3, and 4

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  • $\begingroup$ Thanks for your answer. But why $P^{20}_1$? How does one know that we have 20 men? This is dependent on whether a man or a woman is chosen before the 3rd seat... $\endgroup$ Commented Sep 11, 2017 at 23:10
  • $\begingroup$ So, you're saying that the way we fill the seats is not sequentially, like 1st seat, then 2nd, then 3rd and only then the 4th... $\endgroup$ Commented Sep 11, 2017 at 23:12
  • $\begingroup$ Yes, it doesn't have to be sequential. $\endgroup$
    – Momo
    Commented Sep 11, 2017 at 23:14
  • $\begingroup$ By the way, what if it had to be sequential? Would this still be correct, or would we have to do it with conditional prob.? $\endgroup$ Commented Sep 12, 2017 at 0:00
  • $\begingroup$ Then you should have used conditional probabilities, like in your original solution. $\endgroup$
    – Momo
    Commented Sep 12, 2017 at 0:06

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