Animated Solution for Mathematics - Probability: A fair $n(n>1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $n$ is equal to

Imagine you are standing in a casino, holding a mysterious die with $n$ faces. You are told to roll it repeatedly, but there is a catch: you must keep rolling until you see a number strictly less than $n$.

This is not just a game; it is a fundamental experiment in probability theory known as a 'wait-time' experiment. We are looking for the moment of 'success'—the moment the experiment stops.

To solve this, we must first define our terms. In this experiment, success is defined as rolling any number from the set $\{1,2,\dots ,n-1\}$.

Since there are $n-1$ such numbers, and the die is fair, the probability of success $p$ is simply the number of favorable outcomes divided by the total number of outcomes:

Conversely, failure is rolling the number $n$, which forces us to roll again. There is only $1$ such face, so the probability of failure $q$ is $q=\frac{1}{n}$.

Notice how the sum of probabilities is consistent:

When you repeat independent trials until you achieve your first success, you are walking the path of the Geometric Distribution. This distribution is the mathematical backbone of all 'wait-time' scenarios.

The random variable $X$, representing the number of tosses required, follows this distribution perfectly. One of the most elegant results in probability is the mean of this distribution.

The expected number of trials, $E[X]$, is given by the reciprocal of the probability of success:

Think about it: if success is rare (small $p$), you expect to wait a long time (large $E[X]$). If success is common (large $p$), you expect to wait a short time. The formula $E[X]=\frac{1}{p}$ captures this intuition perfectly.

Now, let us apply this to our specific problem. We know $p=\frac{n-1}{n}$, so the theoretical mean is:

The problem statement gives us a crucial piece of information: the mean is $\frac{n}{9}$. We set our theoretical mean equal to the given value:

Since $n>1$, we know $n eq 0$, so we can safely divide both sides by $n$ to get:

By comparing the denominators, we immediately see that $n-1=9$, which leads us to $n=10$.

We have found that our die has $10$ faces. This problem is a beautiful example of how identifying the underlying distribution—in this case, the Geometric Distribution—can turn a complex-sounding word problem into a simple algebraic equation.

Never fear the 'wait-time' experiment; just find your $p$, take its reciprocal, and the answer will reveal itself.