Question

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?

Answer 1

Answer: (a). prob = 0.1543

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

Step-by-step explanation:

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.