In one of the earliest posts to the blog last year I set a puzzle where I suggested Smartodds were offering employees the chance of increasing their bonus, and you had to decide whether it was in their interests to accept the offer or not.
<They weren’t, and they still aren’t, but let’s play along>.
Same thing this year, but the rules are different. Eligible employees are invited to gamble their bonus at odds of 10-1 based on the outcome of a game. It works like this…
For argument’s sake, let’s suppose there are 100 employees that are entitled to a bonus. They are told they each have the opportunity to increase their bonus by a factor of 10 by playing the following game:
- Each of the employees is randomly assigned a number between 1 and 100.
- Inside a room there are 100 boxes, also labelled 1 to 100.
- 100 cards, numbered individually from 1 to 100, have been randomly placed inside the boxes, so each numbered box contains a card with a unique random number from 1 to 100. For example, box number 1 might contain the card with number 62; box number 2 might contain the card with number 25; and so on.
- Each employee must enter the room, one a a time, and can choose any 50 of the boxes to open. If they find the card with their own number in one of those boxes, they win. Otherwise they lose.
- Though the employees may discuss the game and decide how they will play before they enter the room, they must not convey any information to the other employees after taking their turn.
- The employees cannot rearrange any of the boxes or the cards – so everyone finds the room in the same state when they enter.
- The employees will have their bonus multiplied by 10 if all 100 of them are winners. If there is a single loser, they all end up with zero bonus.
Should the employees accept this game, or should they refuse it and keep their original bonuses? And if they accept to play, should they adopt any particular strategy for playing the game?
Give it some thought and then scroll down for some discussion.
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A good place to start is to calculate the probability that any one employee is a winner. This happens if one of the 50 boxes they open, out of the 100 available, contains the card with their number. Each box is equally likely to contain their number, so you can easily write down the probability that they win. Scroll down again for the answer to this part:
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There are 100 boxes, and the employee selects 50 of them. Each box is equally likely to contain their number, so the probability they find their number in one of the boxes is 50/100 or 1/2.
So that’s the probability that any one employee wins. We now need to calculate the probability that they all win – bearing in mind the rules of the game – and then decide whether the bet is worth taking.
In summary:
- There are 100 employees;
- The probability that any one employee wins their game is 1/2;
- If they all win, their bonuses will all be multiplied by 10;
- If any one of them loses, they all get zero bonus.
Should the employees choose to play or to keep their original bonus? And if they play, is there any particular strategy they should adopt?
If you’d like to send me your answers I’d be really happy to hear from you. If you prefer just to send me a yes/no answer, perhaps just based on your own intuition, I’d be equally happy to get your response, and you can use this form to send the answer in that case.
This is a variant on a puzzle pointed out to me by Fabian. I think it’s a little more tricky than previous puzzles I’ve posted, but it illustrates a specific important statistical issue that I’ll discuss when giving the solution.
