Question: This activity will heighten your awareness of situations that are considered binomial, it will help you determine if you grasped some of the key topics from our study of Chapter 5, and in that way it will help prepare you for answering some of the probability questions covered on Quiz #2 as well as the final exam. Furthermore, this activity meets the course outcome to apply basic concepts of probability in order to assess the likelihood of an event.

Natural gas is used in approximately 60% of American homes. Source: Discovery Channel's Curiosity.com , 2011. Let X be the number of homes that use natural gas, and n be the number of homes you sampled from your state. Assuming this percentage applies to your state ^* , then we have a binomial distribution because we have met the criteria:

1. The outcomes are either success or failure where for this situation, a success is
   S: gas is used in the home
   A failure is, F: gas is not used in the home
2. Letting n be the number of homes in your sample from your state, then we have a fixed number of trials.
3. The trials are independent. Just because one home uses gas does not necessarily mean that the next selected home does.
4. The probability of a success remains constant from trial to trial; for this problem P(S)=60%.

Here's what you should do, rounding all answers to the nearest thousandth:

1. Calculate the expected number of homes in your sample that use natural gas (that is, find the mean) AND calculate the standard deviation of this binomial probability distribution.
2. Calculate the probability that exactly 25 of the homes in your sample use natural gas.
3. Calculate the probability that between 25 and 30 of the homes in your sample use natural gas. Use the binomial formula. (This is part C used in Normal Distribution Activity )
4. Calculate the probability that more than 200 of your n homes use natural gas?
5. If your sample was only comprised of 4 homes, what would be the probability that at least one of them use natural gas? (You could use the binomial tables in the appendix of our textbook, if you so desire, to answer this question.) . (This is part E used in Normal Distribution Activity )

^* Note though that in actuality this percentage should apply to a random selection of homes throughout the USA and that the percentage for individual states might vary somewhat from this figure of 60%.

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