Question 1147272
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The problem specifically asks you to use trial and error to find the answer.  We can't help you with that....<br>
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<br>
({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).<br>
Plug in values of x until you find the smallest value that makes the probability greater than 0.75.<br>
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.<br>
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.