**Question:** You have $7$ marbles in a box, $3$ are white and are numbered from $1$ to $3$ and the other $4$ are grey and are numbered from $4$ to $7$. You extract them one by one till you empty the box. Calculate the following probabilities:

* Probability that the first $3$ marbles extracted are white
* Probability that the first three marbles extracted are white and are also in the sequence $[1, 2, 3]$

**My Approach**  

For the **first part**, I simulated $3$ extractions and I said: first extraction, I have $3$ white marbles out of $7$ total, second extraction I have $2$ out of $6$ and third extraction $1$ out of $5$, so:

$P=\tfrac 3 7\cdot \tfrac 2 6\cdot\tfrac 1 5$

Another idea was using the binomial coefficient: total ways I can extract $3$ marbles out of $7$ is $7\choose 3$, while the way I can extract $3$ white from $3$ total white is $3\choose 3$, so

$$P=\frac{{3\choose 3}}{\binom 7 3}$$

The **second part** took me some troubles. I thought to think this probability as

$\begin{align}P(\text{First 3 are white & in sequence [1,2,3]}) & ={{P(\text{First White and Number 1})}\\\cdot{P(\text{Second White and Number 2})}\\\cdot{P(\text{Third White and Number 3})}}\\[1ex] & =\tfrac 3 7\cdot\tfrac 1 3\cdot\tfrac 2 6\cdot\tfrac 1 3\cdot\tfrac 1 5\cdot\tfrac 1 3\end{align}$.

Are this calculations correct?