$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p_1,\dots, p_n$: $\Mod(S^2\setminus \lbrace p_1, \dots, p_n\rbrace)$. I already know that the mapping class group of the disk $D^2$ minus $n$ points is the braid group $B_n$, whose presentation is well known.
I already know a presentation of $\Mod(S^2\setminus \lbrace p_1, \dots, p_n\rbrace)$ and it can be found for example in the book
Benson Farb, Dan Margalit - A Primer on Mapping Class Groups (Princeton Mathematical)-Princeton University Press (2011), page 258, Chapter 9.
However, I don't understand from there how to get the usual presentation if this group, which is
$$\langle \sigma_1, \dots, \sigma_{n-1} | rel. of B_n, (\sigma_1,\dots, \sigma_{n-1})^n, \sigma_1\dots\sigma_{n-2}\sigma_{n-1}^2\sigma_{n-2}\dots \sigma_1 \rangle. $$
Thus the question is:
How can use the mapping class group $\Mod(D^2)\cong B_n$ to get the above presentation of $S^2\setminus \{p_1,\dots, p_n\}$?
An idea could be also sufficient :).
