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Suppose with staggered entry, $5$ children have come to a teacher to learn alphabets. The teacher assigned them ID according to their arrival, that is, the teacher assigned ID $1$ to the child who came first to her, and then ID $2$ to the next arrival child and so on.

Now the teacher wants to give them prize according to their learning ability. That is, the child who learned the alphabets in the shortest time among the $5$ children will be given the first prize, then the child who learned the alphabets in the second shortest time among the $5$ children will be given the second prize, and so on.

I know that any child can possess any prize. That is, ID $1$ can have the quickest learning ability or can have the 2nd quickest learning ability or he may be the slowest learner among all. The same applies for any ID.

Does the probability that the $i$th arrival possesses the $j$th prize depend on previous $(j-1)$ prizes? That is, can the probability that the 1st arrival gets "the 1st prize" and the probability that the 1st arrival gets "the 3rd prize" be different?

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    Given your comment on `BranchedOut`'s answer I suggest you clarify how the prizes are given. Are they awarded *after* all students have finished learning, on the basis of time to learn (in that case arrival rank is irrelevant)? Or, are they awarded on the basis of who finishes learning first, 2nd, etc (in that case arrival rank is clearly relevant)? – 2017-02-04
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    @A.G. The prizes are given on the basis of time to learn after all students have finished learning. – 2017-02-05

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The way you pose the problem, there is no link between arrival rank (i.e., arrival) and performance. As you mention the prizes depend on time to learn the alphabet, the student who arrives first does not have an advantage (it does not matter that he starts learning before the others).

Now, the only way to distinguish between students would be if you had different expectations of learning times for each student, which you do not.

I would conclude that you are asking a question about probability but you lack information about probabilities.

Therefore the best answer is... we do not know!

  • If you can assume that learning times are identically distributed (i.e., all 5 students are equally good), the the answer is no.

  • You could probably build up exotic examples of 5 students with respective learning time distributions that would lead to a yes answer.

  • You may also build examples where there is a positive or negative correlation between arrival rank and time to learn...

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Yes, you need to exclude the 3rd prize: you can't give the 3rd prize to the 2nd arrival if it's already given away to the 1st arrival. The new probability that the 2nd arrival gets the $j$th prize is $\frac{1}{5-1}=\frac{1}{4}$, unless $j=3$.

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    but 1st arrival can finish his learning after the 2nd arrival. In such case 2nd arrival has still possibility of possessing the 3rd prize also. – 2017-01-26
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    I think I might be a bit confused as to what it's asking then. But if it's already known that the 1st arrival gets the 3rd prize, then I think it would still hold that you should exclude the third prize from the calculations – 2017-01-26
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    I apologize. I have edited the question. – 2017-01-28
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As far as I understood you ask how you should model your problem. Depending on how you do so, the answer to your question can be yes or no.

In my opinion, the time for learning the alphabet should be completely independent of the time of talking to the teacher. For example, you could model the learning time for any student with ID $i$ by an independent uniformly distributed random variable, $X_i$ say. Then, the probability of the first student getting the first prize is $\mathbb P(X_1\le X_2,X_3,X_4,X_5)$. The probability for the second student to obtain the first prize is $\mathbb P(X_2\le X_1,X_3,X_4,X_5)$. By independence and symmetry, they are the same. The same argument can be used to conclude that the probability of obtaining the third place is the same as the probability of becoming first.

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    I haven't understood what is `time of talking to the teacher`? – 2017-02-01
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    Could you please explain `"The same argument can be used to conclude that the probability of obtaining the third place is the same as the probability of becoming first."`? – 2017-02-01
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    By “time of talking to the teacher” I mean the order at which the students arrive at the teacher. – 2017-02-01
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    Concerning your second question: By the argument above, the probability of the first student becoming third is the same as for any other student becomes third. Hence, all these probibilities equal 1/5. The same is true for getting the first or any other rank. – 2017-02-01