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I am just wondering if anyone can help with this question:

A radio station held a competition where contestants were invited to pick a number from $1$ to $50$. If a contestant picked the ‘winning’ number they won a trip to Vegas. The station picked a new ‘winning’ number at random each time a new contestant played the game. The radio station allowed five contestants to play every day over the course of one week.

  1. Compute the probability that the station will have to pay out for exactly one Vegas trip.

I know that there will be 35 people in total who will guess the number but can the 'winning' number be reused or is it without replacement?

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    $\begingroup$ Well, that would be really a question for the radio station. $\endgroup$
    – lulu
    Nov 26, 2015 at 11:16
  • $\begingroup$ Unfortunately I wouldn't be able to ask them in an exam. Thanks for your input though! $\endgroup$
    – luke mcneil
    Nov 26, 2015 at 11:34
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    $\begingroup$ I'm afraid this question doesn't have a correct answer. We could provide an answer for probability with or without replacement, but which interpretation is the correct one is beyond anyone except the presenter of the question. $\endgroup$
    – kviiri
    Nov 26, 2015 at 12:23
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    $\begingroup$ @lukemcneil as kviiri pointed out, the current question does not have an answer. But perhaps you could edit the question to ask what you should do if you came across a exam question with multiple possible interpretations, as that seems to be what you really want to know. $\endgroup$
    – Marc
    Nov 26, 2015 at 14:15

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The only way this makes sense if it is with replacement. That is each contestant can chose any number. Here is why:

  • If it were without replacement it would not make sense to assign a new winning number after each try. This makes sense in a scenario with replacement so that each contestant still has a $1$ in $50$ chance of winning, as earlier attempts do not "eliminate" a number.

  • Somewhat less seriously If it were without replacement and this is on radio it would not be easy to communicate which number are still available.

The above are in my opinion fairly strong reasons for it being with replacement, but in an exam-situation I would still advise to ask for clarification. Now one never knows but I would hope you do get a reply to a to the point question on the interpretation of the question. Do not say something vague like "I do not understand this." but ask specifically like this: "It is not clear to me from the text, can a participant chose the same number as a preceding participant. Which scenario is to be assumed?"

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