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Applications of Probability

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Everyday Uses
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Objective

In this concept you will learn of some of the everyday uses of probability and statistics in the real world. By the end of the lesson, you should see that Probability and Statistics represents one of the most honestly useful math topics you are likely to study.

Concept

What does buying insurance or taking out a loan at the local bank have to do with statistics? What does predicting the weather have to do with probability? Why do boys generally pay more for car insurance than girls?

Stay tuned, after the concept and exercises below, we will return to these questions and discuss the answers.

Watch This

http://youtu.be/NSSoMafbBqQ  Khan Academy - Term Life Insurance and Death Probability

Guidance

Nearly every kind of business can benefit from an application of statistics. By knowing what product to sell at a particular time of year, or to a particular customer, a business can make the best use of product placement and advertising. Knowing what time(s) of the day, week, or year are the busiest, a manager can efficiently schedule her employees so as not to waste labor costs.

One type of business that makes extensive use of statistics is insurance sales. Insurance companies are just like most companies from the standpoint that they are in business to make a profit for their investors. That does not mean that buying insurance is a bad idea for individual people, or that the companies deliberately overcharge their customers, but it does mean that the companies are very careful to charge enough for each policy to ‘insure’ that the company makes money overall.

How can the companies know for certain how many people are going to make claims against their insurance policies? Or how big their claims will be? They can't know for certain since they don’t have a way to see the future, but they can get a very reliable idea of the average number of claims from a specific population of people through the use of sample groups and the application of probability and statistics.

Example A

Suppose a particular insurance company has 100,000 clients, and research suggests that 1 out of each 250 people are likely to make a claim in a given year, for an average of $15,000 per claim. Would a yearly premium of $750 result in a profit for the company, given these statistics?

Solution: Essentially, since 1 out of each 250 people will cost the company $15,000, the total premiums for each 250 people need to be greater than $15,000 for the company to make a profit.

In other words, each policy is responsible for \frac{1}{250}{th}  of $15,000, so each policy needs to cost more than :

\frac{\$150,000}{250}=\$600

Since the company plans to charge $750 per year for each policy, that should result in a profit of $150 per policy on average, for a total estimated profit of  \$150 \times 100,000=\$15,000,000  Not bad!

Of course, that level of profit is only likely if the original research and statistical calculation of $15,000 per 250 people is correct (and the company has no other expenses – which is unlikely), so you can bet that huge multi-million dollar companies are very careful to make sure their calculations are as accurate as possible!

Example B

Predicting the weather is a tricky job. There are a nearly infinite number of possible variables that can affect the temperature and chance of precipitation for any given day. Of course, a weatherman cannot possibly take all of these variables into account every time he/she makes a prediction, so he/she must identify the most influential variables and just watch them closely for each prediction. Suppose that according to records, it has rained an average of 5 days during the month of April for each year over the last 15 years. If it is currently April 25^{th}  and there has been no rain, should the weatherman warn everyone to bring an umbrella to work for the next five days?

Solution: Maybe. However, the information regarding the average number of rainy days in April over the last 15 years probably won't have much to do with it. Although the history may be suggestive of a particular number of rainy days, it is certainly no guarantee of a specific result. If the weather conditions such as temperature, cloud cover, relative humidity, etc., are all conducive to rain, then he/she is likely to predict rain, but the fact that there are only five days left is certainly no assurance that there must be rain all five final days so that the average will be fulfilled.

Example C

Suppose a car insurance company reviews the police records for thousands of speeding tickets and minor car accidents over a ten-year period, and notes the following:

Speeding Tickets “Fender Benders
Boys ages 16 - 23 4,532 1,725
Girls ages 16 - 28 1,242 1,715

Would it make sense for the company to charge the same rates for boys and girls?

Solution: It certainly does not look like it.

According to the statistics, boys are nearly four times as likely to drive over the speed limit, and although there were slightly fewer recorded accidents for girls than boys, note that the age range for the girls was greater than for the boys. The greater age range suggests that there may have been more girls actually driving than boys, yet they ended up in nearly the same number of accidents!

However, it is extremely important to note that without data regarding the actual number of boys and girls in each group, we can't really get a good feel for the overall increased likelihood of boys making claims.

Concept Problem Revisited

What does buying insurance or taking out a loan at the local bank have to do with statistics?

It should make sense now to think that the interest rate you pay for a loan or the premium you pay for insurance is likely based to a great degree on what the statistics say about your likelihood to pay the loan off in a timely manner or make a claim against your policy.

What does predicting the weather have to do with probability?

Now we know that there are a number of different variables associated with the weather, including: historical weather patterns, current temperature and local trends, current humidity, regional weather patterns, and many more. The greater the number of variables taken into account and the more accurate the calculations, the more likely it will be that a particular weather prediction will be correct.

Why do boys generally pay more for car insurance than girls?

Statistically, boys drive faster than girls, and get into more accidents. That does not mean, of course, that any particular driver is more of a risk than any other just based on his or her sex, but overall it does mean that it makes logical sense for an insurance to charge a premium for male teen drivers.

Vocabulary

A sample is a smaller portion of all of the possible members of a given set. Chosen properly, a sample can provide a good approximation of the overall behavior of the entire group.

Data can be either singular or plural, and refers to pieces of information gathered from a sample for the use of statistical calculation.

In the context of statistics, a variable is a characteristic of the elements in your data set.

Guided Practice

  1. Which of these number(s) cannot represent a probability?
    1. -0.00001
    2. 0.5
    3. 1.001
    4. 0
    5. 1
    6. 20%
  2. Which types of studies below might a retail store make use of to improve sales?
    1. The average amount of money spent by customers of various age groups
    2. The type of products preferred by customers of various age groups
    3. The best and worst selling products to female and male customers
    4. The busiest season for selling a particular product
    5. All of the above
  3. How could an understanding of statistics benefit you on university entrance exams, such as ACT or SAT?

Solutions:

  1. A probability can only be between 0% and 100% (or between 0 and 1, as a decimal). 0% means it will not happen, and 100% means it will happen, every number between represents some shade of “it may happen”. Choice “a” is negative, and choice “c” is greater than 100%, so neither is possible.
  2. “e” is correct. All of those bits of information would be very useful to a skilled retail manager, owner or salesperson (particularly if paid on commission).
  3. SAT/ACT preparation courses make extensive use of statistics to help students understand when to expect a problem to be easy or hard, when it is worth spending extra time solving a particular question, and when it is not.

Practice

  1. What is the difference between probability and statistics?
  2. What are three industries that make use of statistics?
  3. Why do girls generally pay less for car insurance?
  4. Why do retail stores start carrying holiday decorations and promoting gifts well before the holiday season?
  5. When an animated film is played in theaters, why is it often preceded by previews of children’s movies?
  6. Why are commercials for toys played during cartoons?
  7. Why are banner ads for beauty magazines often displayed on Hollywood rumor-type web pages?
  8. Why are children’s athletic shoes often promoted by professional sports players?

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