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TheMathNoob

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**1. The problem statement The United States Senate contains two senators from each of the 50 states. (a) If a committee of eight senators is selected at 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 probability that a group of 50 senators selected at random will contain one senator from each state?**

## Homework Equations

Combinatorial Methods

## The Attempt at a Solution

I see this problem like trying to determine the probability that at least one senator from California will be in the committee, so the number of ways how this can happen is 1 or 2 senators are in the committee, so it would be:

Number of outcomes of the given event=(99 choose 7)+ (98 choose 6)

Sample space= 100 choose 8

Assuming that one senator is already in the committee then what is left to do is to place the remaining 7 from the 99 available senators. This would the number of outcomes in the case in which there is just one senator. And assuming in the next case that there are 2 senators from the same state then we just place other 6 senators from the 98 available senators.

P(N)= ( (99 choose 7)+(98 choose 6))/100 choose 8

For part B, I imagine this like making little groups of 1 senators from a set of 2 senators from each state, so the number of outcomes would 2^50 and the sample space would be the same 100 choose 8, so the probability is the ratio of what was mentioned.

How can I increase my ability to detect mistakes?. When I do a math problem, I can't see my mistake and then I look at the solution and realized that it was a very stupid mistake which was very easy to see. I can do and study my hw, but what if I try every problem without getting the right answer?. I check the answer key and realize that it was easy and I feel like I understand it, but then it keeps happening all the time. How can I be more accurate in knowledge?

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