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Luck and bad luck in the raffle

2022-05-22T04:52:24.767Z

The company is giving away a fancy tablet – as well as wine and towels. The distribution of profits is surprising. What does that reveal about the number of wine bottles?



Photo:

THE MIRROR

The annual company raffle is very popular with employees.

After all, there are no rivets - everyone can take home a prize.

This year the main prize is a tablet.

The head of human resources provided towels and wine as additional prizes.

The number of prizes available corresponds exactly to the number of employees.

So the prize for each employee is either the tablet, a towel or a bottle of wine.

All employees take part in the raffle, even if one or the other gets angry about the towels as prizes.

Why wasn't there wine for all the tablet non-winners?

Helene draws the first ticket and is lucky: she wins the tablet.

Her colleague Ahmed draws the second lot and gets a towel.

The probability of such a distribution is 10 percent.

How many wine bottles were raffled?

Only two wine bottles were raffled.

At first glance, the task hardly seems solvable.

But after some reflection, it turns out that the company must be quite small and cannot have more than ten employees.

And that only a certain number of towels and wine bottles to be raffled off are possible, so that the probability of the constellation described is one tenth.

Let's start by calculating the probability that Helene will win the tablet and Ahmed will win the towel.

If

n

is the number of participants in the

raffle

and

m

is the

number of towels

, the following equation applies:

p = 1/10 = 1/n * m/(n-1)

Explanation: Helene wins the tablet with p=1/n.

Ahmed wins one of the m towels with p=m/(n-1) if Helene has not won a towel.

n must be at least four, because we are talking about a tablet, towels in the plural and wine - so at least one bottle.

The fraction m/(n-1) in the equation above is therefore always less than 1.

It follows that n must be less than 10.

Otherwise the product 1/n * m/(n-1) cannot be 1/10 as required.

We transform the equation 1/10 = 1/n * m/(n-1) so that it no longer contains any fractions:

n*(n-1) = 10*m

Now the prime number decomposition helps us further.

The prime factor on the right is 5. Because n is between 4 and 9, n or n-1 must equal 5 for the prime factor 5 to also appear on the left.

We find two different solutions:

n = 5 --> m = 2

n = 6 --> m = 3

nm-1 is the number of wine bottles.

This number is 2 in both cases. So there are only two bottles of wine and either five or six people work in the company.

I discovered this amazing tombola puzzle in Frank Schwellinger's book »Das Neue Haus vom Nikolaus«.

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Source: spiegel

All business articles on 2022-05-22

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