Geometric distribution not quite right

Question

The number of positive ions crossing a detector in a certain time until the first negative ion is seen is a random variable with $E(x) = 11$. Each cross is an independent event. The distribution function is given by $F_X (X \leq x) = 1 - (1-p)^x$ with $x = 1, 2, 3, ...$.

What is the value of $p$? What is the probability of 4 consecutive positive ions crossings until you see the first negative ion crossing?

I believe this is a (binomial) geometric distribution with $X =$ "number of positive ions crossing a detector in a certain time until the first negative ion is seen" and for wich $E(X) = \frac{1}{p} = 11 \iff p = \frac{1}{11} = .090$.

Now that I have $p$ I believe that I may have to calculate $F(X \geq 5)$ but then I do $$F_X(X \geq x) = 1- F_X(X \leq x) = 1 -(1-(1-p)^x) = (1-p)^x = (1-.090)^5 = .624$$ this doesn't seem quite right to me because it is too high. Am I thinking something wrong here?

Answer 1

Your interpretation is correct, even the correct calculation (see @callculus latest remark) should be $P(X\ge 5)=(1-0.090)^4$. Your distribution is a geometric, not a binomial, with support $x=1.2.3,\dots$.

What is the probability of 4 consecutive positive ions crossings until you see the first negative ion crossing?

if the question was

What is the probability of exactly 4 consecutive positive ions crossings until you see the first negative ion crossing?

the answer was lower:

$$\left(\frac{10}{11}\right)^4\cdot \frac{1}{11}=6.21\%$$

Answer 2

What is the probability of 4 consecutive positive ions crossings until you see the first negative ion crossing

This probability you can calculate with the $\textrm{pmf}$, not cdf. The $\textrm{pmf}$ is

$$f_X(k)=(1-p)^{k-1}\cdot p, \ \ \forall \ k\in \{1,2,3,...\}$$

So you need the probability for $k=5$.

Remark $(\textrm{11/9/2021})$

$F_X(X \geq x) = 1- F_X(X \leq x)$ is not right, since $X$ is a $\underline{\color{green}{\textrm{discrete}}}$ random variable. In this case we have

$$F_X(X \geq x) = 1- F_X(X < x)=1- F_X(X \leq x-1)$$