# Ten Coins

We lately discussed a simple looking question, which seems to be more difficult that expected.

You have $10$ coins numbered $1$ through $10$ on one side and with a $0$ on the other side. You toss all coins and sum up the values that face up. What is the probability that the sum is greater than or equal to $45$?

We found two ways to answer the question. But we are not satisfied with our solutions because you cannot(?) use them for a more general version of the problem. Maybe you have better solutions?

1. solution:

Because there are 10 (perfect, not biased) coins, there are ${ 2 }^{ 10 }$ possible outcomes. In order to get the probability, we need to know for how many outcomes the sum is greater than or equal to $45$. If the sum of the sides facing up is greater than or equal to $45$ then the sum of the sides facing down is less than or equal to $10$. Therefore we just need to know for how many outcomes the sum of the sides facing down is less than or equal to $10$. Or in other words: How many ways are there to write a natural number less than $11$ as the sum of distinct natural numbers.

You can find all possibilitys manually or just ask OEIS: There are $1 + 1 + 1 + 2 + 2 + 3 + 4 + 5+ 6+8+10=43$ ways to write a number less than $11$ as sum of distinct natural numbers. (There is $1$ way for $0$, when all zeros facing down; there is $1$ way to write $1$ as sum of distinct natural numbers; $1$ way for $2$; etc.)

Hence the probability is $\frac { 43 }{ { 2 }^{ 10 } } \approx 0.042$

2. solution:

We made a few python simulations and figured out that it is a normal distribution with $\mu \approx 27.5$ and $\sigma \approx 9.8$. By taking the area under the curve from $44.5$ to $\infty$ with the given mean and standard deviation we got a probability of $\approx 0.0414$.

But none of these two methods work for a more general question:

You have $n$ coins numbered $1$ through $n$ on one side and with a $0$ on the other side. You toss all coins and sum up the values that face up. What is the probability that the sum is greater than or equal to $x$?

How would you solve this general version of the question?

Note by CodeCrafter 1
2 weeks, 6 days ago

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