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Math Help - A real-life question regarding probabilities

  1. #1
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    A real-life question regarding probabilities

    I have a real-life math question regarding probabilities. My daughter works for a large bank and is in charge of the committee that is planning a holiday event for all the employees. This event will include a raffle.

    Here are the details:
    Number of attendees: 123
    Number of prizes: 26
    Number of tickets: 5 for each employee = 123 x 5 = 615 tickets


    The individual prizes will be displayed on tables next to a container that will hold the tickets. Each attendee may place one or more of their allotted five tickets into any prize container. Obviously, the more tickets one places in a single prize container, the greater the chances of winning that prize.


    When the actual drawing occurs, a committee member will randomly select a single ticket from the ticket container for that particular prize. The winner will be announced and the next drawing will proceed. Each of the 26 drawings will occur sequentially until the last prize is awarded.


    There is some dissension within the committee as to whether a single attendee will be allowed to win more than one prize. To help with this decision, I would like to calculate the chances of any one individual's ticket(s) being drawn more than once.


    Based on the numbers above, can someone help me determine the probability of any one single person winning more than one prize?
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  2. #2
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    Quote Originally Posted by Shadoefax View Post
    I have a real-life math question regarding probabilities. My daughter works for a large bank and is in charge of the committee that is planning a holiday event for all the employees. This event will include a raffle.

    Here are the details:
    Number of attendees: 123
    Number of prizes: 26
    Number of tickets: 5 for each employee = 123 x 5 = 615 tickets


    The individual prizes will be displayed on tables next to a container that will hold the tickets. Each attendee may place one or more of their allotted five tickets into any prize container. Obviously, the more tickets one places in a single prize container, the greater the chances of winning that prize.


    When the actual drawing occurs, a committee member will randomly select a single ticket from the ticket container for that particular prize. The winner will be announced and the next drawing will proceed. Each of the 26 drawings will occur sequentially until the last prize is awarded.


    There is some dissension within the committee as to whether a single attendee will be allowed to win more than one prize. To help with this decision, I would like to calculate the chances of any one individual's ticket(s) being drawn more than once.


    Based on the numbers above, can someone help me determine the probability of any one single person winning more than one prize?
    Since you don't know how the tickets are placed there will be no answer to this question, we would have to model the ticket placing strategies of all the player to do so.

    I also don't see the point, why do you not want to treat the tickets equally, also there is the problem of a player placing one ticket against a prize that will be drawn early and 4 tickets against a prize that will be drawn later. It would be grossly unfair if winning the earlier prize precluded their winning the later prize which they obviously would prefer.

    CB
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    Thank you CapianBlack for your response.

    Quote Originally Posted by CaptainBlack View Post
    I also don't see the point, why do you not want to treat the tickets equally ...
    By giving each person five tickets, they have the option of dividing their chances of winning any particular prize according to how desirable that prize is to them.

    I agree that this is a virtually unsolvable problem. I would imagine one would have to determine all possible combinations of ticket distribution to create a working sample space to calculate the probabilities. And it would seem that the probabilities would change after each drawing.

    Above all, my daughter wants to keep the raffle as fair as possible. This brings up some other questions. If the ticket containers are transparent (e.g., a fishbowl) then everyone can see which items are most popular by examining the number of tickets in the container. Does this give the last person distributing his/her tickets a greater advantage than the first person? The first person will see 26 empty fishbowls and the last person may see 100 tickets in the diamond earrings fishbowl but only a few tickets in the one belonging to the "slap chop" food processor. Would this knowledge afford the last person some information that could be used to determine a way to maximize their chances of winning that (or any other) prize? If so, then it would seem that each successive "bidder" could have an unfair advantage over the preceding bidder (assuming they would know how to apply this knowledge). If using fishbowls does provide the possibility of maximizing latter bidder's chances then we could use some sort of opaque container to keep the tickets hidden. But keeping them visible is preferred as it not only makes the raffle more exciting (by offering at least a perceived ability to affect the chances of winning), but also reduces any sense of impropriety in the process.

    Another alternative would be to give each winner the option of declining the prize in order to keep open the possibility of winning another, more desirable one. This "one-in-the-hand" vs. "two-in-the-bush" dilemma could also add to the excitement.

    One final question comes to mind would knowing the order in which the items are awarded offer any strategic advantage when deciding how a person might distribute their tickets? For instance, if knowing that the drawings were to be held for items in descending (or ascending) retail value, is there a way to devise a best-case strategy of winning any particular item (possibly by varying the way one's tickets are distributed)?
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    I agree that this is a virtually unsolvable problem. I would imagine one would have to determine all possible combinations of ticket distribution to create a working sample space to calculate the probabilities. And it would seem that the probabilities would change after each drawing.
    I'm not sure if the probabilities would change after each drawing since the draws would be independent. Obviously transferring losing tickets from the first bowl would be hideously unfair hence the number of tickets in subsequent bowls will not change after the first draw. For example if box 2 had 108 tickets in when box 1 is drawn then 108 would remain of which one is a winner.

    Above all, my daughter wants to keep the raffle as fair as possible. This brings up some other questions. If the ticket containers are transparent (e.g., a fishbowl) then everyone can see which items are most popular by examining the number of tickets in the container. Does this give the last person distributing his/her tickets a greater advantage than the first person? The first person will see 26 empty fishbowls and the last person may see 100 tickets in the diamond earrings fishbowl but only a few tickets in the one belonging to the "slap chop" food processor. Would this knowledge afford the last person some information that could be used to determine a way to maximize their chances of winning that (or any other) prize? If so, then it would seem that each successive "bidder" could have an unfair advantage over the preceding bidder (assuming they would know how to apply this knowledge). If using fishbowls does provide the possibility of maximizing latter bidder's chances then we could use some sort of opaque container to keep the tickets hidden. But keeping them visible is preferred as it not only makes the raffle more exciting (by offering at least a perceived ability to affect the chances of winning), but also reduces any sense of impropriety in the process.
    I would go with a sealed, opaque box, not unlike a ballot box at elections into which contestants put their tickets in. The key would be held by either your daughter/a committee member/someone not involved in the draw and not opened until the draw is to take place and then infront of everyone.
    The entrants would need to have trust in whoever has the key but I see no other realistic way, especially if the box is opened infront of everyone.


    Another alternative would be to give each winner the option of declining the prize in order to keep open the possibility of winning another, more desirable one. This "one-in-the-hand" vs. "two-in-the-bush" dilemma could also add to the excitement.
    Don't agree with this. If one person wins all 5 prizes (certainly possible) then they're a lucky sod but it would be more improper to take the prizes from them.

    One final question comes to mind … would knowing the order in which the items are awarded offer any strategic advantage when deciding how a person might distribute their tickets? For instance, if knowing that the drawings were to be held for items in descending (or ascending) retail value, is there a way to devise a best-case strategy of winning any particular item (possibly by varying the way one's tickets are distributed)?
    I think it's quite clear there would be an advantage. If you know the car is going to be drawn last and the blender first and you also know the order they'll be drawn you'll get more going for the car. The best strategy for winning something is to go for the lowest value prize you would like.
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