## The Probability behind Product Warranties

### Topic

The Probability behind Product Warranties

### Vocabulary

- Independent events
- Dependent events

### Student Exploration

#### How do companies determine the length of the warranties for their product? What are the chances that if your product breaks, it will still be covered by the warranty?

Have you bought an item recently that had a warranty? Frequently cars, household appliances, electronics, and watches have product warranties. Many companies offer warranties as they give customers a sense of security that if the product breaks, they will be able to get it replaced. However, what are the chances that if a product break, it will be covered by the warranty?Nearly every major company in the manufacturing industry(companies that make products) does thorough testing to determine what could cause their product to break or fail. They also do thorough testing to determine after regular use, how long will the product last. Then the length of a product’s warranty is determined based on the probability that the product will fail. Therefore a product’s warranty length and failure rate are dependent events, as the probability of one event affects the probability of the other event.

So, let’s get back to the original question, “What are the chances that if your product breaks, it will still be covered by the warranty?” For this question in \begin{align*}P(A \ and \ B) = P(A) P(B|A), P(A)\end{align*} is defined as the probability that the item will break and \begin{align*}P(B|A)\end{align*} as the probability that the warranty is still valid when the item breaks. Your task is to choose some items that you have bought recently find their warranties and then research their product failure rates. Unfortunately, companies do not publish the results of their product testing as they are not required to and as it wouldn’t be a good strategy to increase sales, so you will have to do some serious research to find this information. Below are some examples of websites with some product failure rates and other information.

- See “Exam P Practice Problem 6a” for an example of Conditional Probability being used for to calculate insurance coverage. http://probabilityexam.wordpress.com/tag/exponential-distribution/

http://www.consumerreports.org/cro/index.htm

http://www.squaretrade.com/htm/pop/lm_failureRates.html

http://arstechnica.com/gaming/2008/02/xbox-360-failure-rates-worse-than-most-consumer-electornics/

http://www.consumer.org.nz/reports/unsafe-kids-products/failure-rate

http://www.digitaltrends.com/mobile/android-hardware-failure-rate-higher-than-iphone-and-blackberry/

http://www.consumerreports.org/cro/search.htm?query=iphone

Electronic items often have a warranty.

### Resources Cited

http://www.consumerreports.org/cro/index.htm

http://www.squaretrade.com/htm/pop/lm_failureRates.html

http://arstechnica.com/gaming/2008/02/xbox-360-failure-rates-worse-than-most-consumer-electornics/

http://www.consumer.org.nz/reports/unsafe-kids-products/failure-rate

http://www.digitaltrends.com/mobile/android-hardware-failure-rate-higher-than-iphone-and-blackberry/

http://www.consumerreports.org/cro/search.htm?query=iphone

http://probabilityexam.wordpress.com/tag/exponential-distribution/

### Connections to other CK-12 Subject Areas

- Independent Events and Sample Spaces
- Conditional Probability
- Theoretical and Experimental Probability

Images from: http://commons.wikimedia.org/wiki/File:Consumer_img02.jpg

http://pixabay.com/en/computer-black-notebook-top-icon-33521/

http://pixabay.com/en/computer-photo-digital-camera-lense-36019/