Politician probablity help

[rdn98]

A committee of 50 politicians is to be chosen from among our 100 senators. If the selection is done at random, what is the probability that each state will be represented?


My friend tried to help me out, and this is what he came up with.

Since there are 2 senators from each state, you got to figure out the proablity of that into the calcuations. Figure out the proability of picking a senator from each state, add them together, then multiply it by 100C50.


100C50 * sum [2/(100-i)]
where i goes from 0 to 49

the summation part is the probablity of the senators being picked from each state

When I do the math, I get a huge answer, and I can't have the probabilty bigger than one. So I did 100C50 divided by the huge answer, and I get .7265 which could be correct.

Frankly, I don't understand the logic to solving this problem. Can someone help me out?

 

[robert Ihnot]

I would say that one must fill fifty slots with 2 choices each. The total probability as you realize is 100C50. The answer I can get from the use of a downloaded calculator: http://pari.math.u-bordeaux.fr/pub/pari/manuals/users.pdf Or use Sterling's Formula, which simplifies somewhat.

 

[HallsofIvy]

There are ₁₀₀C₅₀ ways of choosing 50 people from a group of 100. The set of all possible "events" has ₁₀₀C₅₀ members.
A "success" in this problem would be having exactly one senator from each state.
How many ways are there of choosing exactly one senator from each state? Well, there are there are 2 ways of choosing a senator from Maine, 2 ways of choosing a senator from Vermont,2 ways of choosing a senator from New Hampshire, etc. That is, a total of 2⁵⁰ ways of choosing one of the two senators from each of the 50 states. Since 2⁵⁰ of of the ₁₀₀C₅₀ events are "successes", the probability of a success is 2⁵⁰/₁₀₀C₅₀ . I don't think that comes out anywhere near your answer.