Negative binomial with conditional probability

Question

Let X be a random variable that follows a negative binomial distribution: NB(r=4, p=0.4) Calculate P(X = 8 | x > 6)

I know how to calculate P(X = 8):

$$ \binom{7}{3} \cdot (1 - 0.4)^{7-3} \cdot (0.4)^{4} = 0.1161 $$

And this is what I wrote for P(X > 6), tough I'm not so sure about it:

$$ 1 - P(X \lt 6) = 1 - \big[ P(X = 4) + P(X = 5) \big] $$

But after that I'm stuck. I don't understand how to solve P(X = 8) when there is (P > 6) as a conditional probability.

Thanks!

Answer 1

Just apply the definition.

$\begin{align} \mathsf P(X=8\mid X>6) &= \dfrac{\mathsf P(X=8 \cap X>6)}{\mathsf P(X>6)} \\[1ex] &=\dfrac{\mathsf P(X=8)}{\mathsf P(X>6)} \end{align}$