Probability each state of USA is represented in a committee

Each state of USA has $2$ senators. The committee has $50$ sits. What's the probability that all states has represented in committee?

I know why $\Omega=\binom{100}{50}$, but I can't understand why I have $2^{50}$ probabilities to have all states represented.

I can't figure out how many possibilities there are if $1$ senator is in committee. I don't know where to start.

• The numerator gives the number of ways to pick a committee with exactly one senator from each state. The denominator gives the number of ways to choose 50 senators from a group of 100 senators, i.e., the total number of ways to form the committee. – David Jul 25 '17 at 13:38
• It's a simple application of $\text{probability} = \frac{\text{favourable}}{\text{possible}}$ – Arthur Jul 25 '17 at 13:38

By yourself and with the other answers' help you seem to have figured out most of it, but the sticking point is the $2^{50}$.

You've got 50 states and 50 seats, so in order to have all represented you can look at a seat and say that it's the Texas seat (Texas needs to have a seat somewhere, but if it had 2 then probably Nevada wouldn't get a seat). Then there's 2 options for who sits there. Then for each of the other states there are 2 options, so there are $2 \times 2\times \ ...\ \times2 = 2^{50}$ overall possibilities. If each state had three senators, the answer would be $3^{50}$.

If you don't understand why the 2's multiply together I can explain that too, though that's sort of the first thing you might do in probability so hopefully it makes sense.

• Oh, okay! Now I understood everything! You don't need to explain anything more! Thank you for your answer! – Dear An Jul 25 '17 at 15:05
• No problemo, have fun – George Moore Jul 25 '17 at 15:11

How many ways are there of choosing 50 senators out of 100? The committee must have 50 senators, and there are 100 to choose from.

How many ways are there of choosing 1 out of 2 senators for each of 50 states? You want the probability each state has 1 senator in the committee.

HINT: Recall that the probability of an event $E \in S$, where $S$ is the sample space, is given by $$P(E)=\frac{\operatorname{number of elements in E}}{\operatorname{number of elements in S}}$$ Of course, the total number of ways to distribute $100$ reps among $50$ spots is $$\binom{100}{50}$$ ...but how many ways are there of doing this such that each state gets represented? Can you figure this out?
• No, I can't! I modified my question saying I know what are you saying. I can't figure out why I have $2^{50}$ possibilities of representing each state. – Dear An Jul 25 '17 at 13:55