Dependent Probability from Probability Density Function of a Discrete Random Variable

Question

The number of people that win prizes more than 10.000\$ in a day follow a Discrete Random Variable with Probability Density Function:

$ f(x) = \begin{Bmatrix} ax & x=0,1,2,3,4,5,6,7,8 \\ a(25-x) & x=9,10,11,12,13 \end{Bmatrix} $

Find the probability at least 5 people will win 10.000\$ if it is know that at least 3 people are winning 10.000\$.

I found that: $ a = \frac{1}{106} $

so if I am correct the asked probability is the dependent probability:

$ \frac{P(X \geqslant 5 \cap X \geqslant 3)}{P(X \geqslant 3)} $

but the textbook says $ \frac{P(X \geqslant 5)}{P(X \geqslant 3)} $ which is solved as

$ \frac{1 - P(X < 5 )}{1 - P(X < 3)} = \frac{ \frac{96}{106}}{\frac{103}{106}} = \frac{96}{103}$

Is the joint removed because $ X \geqslant 5 $ contains $ X \geqslant 3 $ ?

Answer 1

It's the other way round. $X\geq 3$ contains $X\geq 5$. The intersection is $X\geq 5$

It can be seen easily if you use a number line, like below: