SOLUTION: A senate committee of five members is to be selected from N Labor and five Liberal senators. Use trial and error to find the minimum value of N, given that the probability of Labor

Question 1147272: A senate committee of five members is to be selected from N Labor and five Liberal senators. Use trial and error to find the minimum value of N, given that the probability of Labor having a majority on the committee is greater than 3/4 .
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The problem specifically asks you to use trial and error to find the answer. We can't help you with that....

If the number of Labor senators is x, then the probability of having a majority of labor senators on the 5-member committee (that is, 3 or 4 o 5 labor senators out of the 5 total) is

({x choose 3) times (5 choose 2)) plus ((x choose 4) times (5 choose 1)) plus ((x choose 5) times (5 choose 0)), and then all that divided by ((x+5) choose 5).

Plug in values of x until you find the smallest value that makes the probability greater than 0.75.

It should make sense to you that if x=5 (the number of Labor and Liberal senators is the same), then there is an equal probability for either party to have a majority on the 5-member committee; so the answer is going to be a number greater than 5.

If you have a good graphing calculator, you can make it do nearly all the work for you. On my TI-83 I am able to enter the complete expression shown above as a function; I couldn't get it to graph the function, but the TABLE option made it easy to quickly see the answer to the question.
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