# Good vs Bad Trial in Probability

I am watching this video on Probability, around the end there is a problem to be solved:

I can't understand how a summation which extends from 1 to infinity can be equal to $$\frac{1}{1-4/6}$$

• Search Geometric series on internet. Commented Nov 8, 2018 at 6:13
• en.wikipedia.org/wiki/Geometric_series#Formula -- it even shows how to derive it. Commented Nov 8, 2018 at 6:46

The series $$S = (\frac 46)^0\frac 16 +(\frac 46)^1\frac 16+... = ar^0 +ar^1 + ar^2...$$ is a geometric series. You can search Geometric Series on internet to get the answer. In addition, you may get the result for the sum of the first n terms of a geometric series, which is $$a \frac {1-r^n}{1-r}$$ and also useful in statistics.
You know the sum is finite, since it is a probability and so must be between $$0$$ and $$1$$. Let's call it $$S$$ where $$S=\frac16+\left(\frac46\right)\frac16 +\left(\frac46\right)^2\frac16 +\left(\frac46\right)^3\frac16 +\cdots$$
Multiplying through by $$\frac46$$ gives $$\frac46 S=\left(\frac46\right)\frac16 +\left(\frac46\right)^2\frac16 +\left(\frac46\right)^3\frac16+\left(\frac46\right)^4\frac16 +\cdots$$
and taking the difference gives $$\dfrac26 S=\dfrac16$$ i.e. $$S=\dfrac12$$, as you might intuitively expect from the symmetry in the question. This is a geometric series