A poll is given, showing 30% are in favor of a new building project.

If 10 people are chosen at random, what is the probability that exactly 2 of them favor the new building project?
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The possible outcomes of selecting 10 people from a population, in which 30% support a building project while 70% do not, include 10 people who all support it and 10 people who all reject it, and everything in between. If exactly two people support it then 8 reject it. That means we have a probability of (0.3)²(0.7)⁸=0.00519 approx multiplied by as many ways we can pick 2 people out of 10. There are 10 ways to pick one out of ten, leaving 9, so there are 10×9=90 ways of picking two people in a particular order, and since there are 2 ways of arranging 2 people, there are 90/2=45 combinations. So we multiply 0.00519 by 45=0.2335 approx. The probability in total for exactly 2 out of 10 supporting the project is 23.35% approx.

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