Elizabeth was just a small-town girl living in a lonely world. She was taking the midnight train going anywhere, anywhere of course being home since she had just got off work and was desperately trying to get home and go to sleep. When it comes to the train station, Elizabeth had the choice between two trains the express, that would take her straight home and the regular that had a few stops in between. She estimated that she had a 27% chance of getting the regular instead of the express. Why wouldn’t she get the express, well it has a larger chance of breaking down mid journey, in fact it has a 52% chance of breaking down, while the regular only has a 6% chance of breaking down. Use this to determine the following.
1) What is the probability that she will get on any train that will break down?
2) What is the probability that she will get on the express train and it will not breakdown
3) Given that the train she was on has broken down, what is the probability that she was on the express train?
4) What is the probability that she will get on a train and it will not break down?
She gets home and begins to wind down for the night, but she may have to do a few things before she goes to bed. She her boyfriend sometimes forgets to do things around the house and they have a cat and a dog. There is a 34% chance that when she arrives home, she will have to walk her dog, and a 52% chance that she will have to feed the cat. There is a 20% chance she will have to do both.
5) What is the probability that she will have to just walk the dog?
6) What is the probability that she will have to just feed the cat?
7) What is the probability that will walk the dog, feed the cat or do both?
8) What is the probability that she will do neither?
So, all the chores are done, and she is ready for bed. However, she lives in an area with a weak electrical structure and there is a 15% chance that her power will go out and as a result she will have to call the electric company and wait for them to come out and fix the problem.
9) Explain how this situation can be used as a Bernoulli trial?
10) What is the probability that the electricity doesn’t go out until the 6 night?
11) Out of 8 nights, what is the probability that the electricity will go out on 5 of those nights?
In most cases when the lights go out it a minor problem needing only $15 to repair the plan, this happens 55% of the time. About 40% of the time the electrician must do a little bit more work do weather and it cost Elizabeth about $30 to fix the electricity. Very rarely, the electricity problem is severe, and the electrician must dig into the ground. This costs Elizabeth about a $100 to repair.
12) What is Elizabeth expected repair cost?
13) What is the standard deviation for the cost?
14) Suppose that the electrician must add $5 to all the prices, what is the new expected cost and standard deviation?
15) Suppose that the normal electrician can make it and one from far away decides to come out but must triple to prices to cover costs. What is the new expected value and standard deviation?
16) So, Elizabeth is working through school right now and she is planning on being an attorney. She does a little research and she finds that out of 1000 possible attorneys about 56% make it all the way through school. Determine a 95% confidence interval to determine the true proportion of people who finish school. Interpret it in words
17) What is the margin of error for the confidence interval in question #16?
18) After some more research, Elizabeth determines that 76% of all attorneys get jobs after the graduate with their degrees. She finds a study on 3000 random attorneys that says that 78% of attorneys are hired after they finish their degree. Is this enough information to say that the percentage of attorneys is greater then what was though before?
19) For the situation in #18 what is the only type of error that can occur?
20) Interpret your error from #19 in words
Bonus Time
1) Considering the situation with Elizabeth cat and dog, what is the probability that she doesn’t have to walk her dog?
2) What does the Central limit theorem tell me?
3) Suppose we want to determine out of the next 6 years, (2191 days), what is the probability that 325 days or less will have electrical problems.
4) You are a contestant in a game show. You are brought up on stage and there are three curtains. Behind one of the curtains is a brand-new car. On curtain #1 there is a message that says the car is behind this curtain, on curtain #2 there is a message that says the car is not behind this curtain, and behind curtain #3 a message says that the car is not behind curtain #1. The host tells you must pick one curtain and that only one of the curtains is telling the truth and the other two are the complete opposite. Using this information, behind what curtain is the car? You must explain why you think this, you cannot just choose a curtain.