The United States Senate contains two senators from each of the 50 states. (a) If a committee of eight senators is selectedat random, what is the probability that it will contain at least one of the two senators from a certain speciﬁed state? (b)What is the probabiﬁty that a group of 50 senators selected at random will contain one senator from each state?

The information in the question tells us that the United States contains two senators from each of the 50 states, this means there is a total of a 100 senators in the senate.

(a). this question tells us to find the probability that it will contain one of the the two senators from a certain specified state.

Explanation:

Consider selecting a committee of 8 senators at random,

the total number of ways to select 8 senators from the Senate of 100 senators is \left[\begin{array}{ccc}100\\8\end{array}\right].

Next is to consider an event Q that at least one of the senators from the specified state is added to the senate,

Let us call the two senators F and G.

where F is the number of possible ways to select 7 members from 98 senetors; \left[\begin{array}{ccc}98\\7\end{array}\right].

Also, G is the number of possible ways to select 6 members from 98 senators; \left[\begin{array}{ccc}98\\6\end{array}\right].

The probability that committee of 8 senators will contain at least one of the two senators from a certain specified state is given thus;

Prob (Q) = \left[\begin{array}{ccc}2\\1\end{array}\right]\left[\begin{array}{ccc}98\\7\end{array}\right] / \left[\begin{array}{ccc}100\\8\end{array}\right] + \left[\begin{array}{ccc}2\\2\end{array}\right] \left[\begin{array}{ccc}98\\6\end{array}\right] / \left[\begin{array}{ccc}100\\8\end{array}\right]

Prob (Q) = 0.1487 + 0.0057 = 0.1543

Prob (Q) = 0.1543

(b). the questions tells us to find the probability it will contain one senator each from each of the states.

consider that a group of 50 senators is selected in random, the total number of ways to select 50 senators from the senate of 100 senators is

\left[\begin{array}{ccc}100\\50\end{array}\right].

Now let us consider that an event Q, that a total of 50 groups contains exactly one senator from each state. for any state, there are two ways for exactly one senator to be in a group of 50.

∴ the probability becomes;

Pr (Q) = 2⁵⁰/ \left[\begin{array}{ccc}100\\50\end{array}\right].

cheers, i hope this helps.