3.4 Contingency tables and probability trees  (Page 4/6)

Try it

In a standard deck, there are 52 cards. Twelve cards are face cards ( F ) and 40 cards are not face cards ( N ). Draw two cards, one at a time, without replacement. The tree diagram is labeled with all possible probabilities.

1. Find $P(\mathrm{FN}\cup \mathrm{NF})$ .
2. Find P ( N $|$ F ).
3. Find P (at most one face card).
   Hint: "At most one face card" means zero or one face card.
4. Find P (at least on face card).
   Hint: "At least one face card" means one or two face cards.

1. $P(\mathrm{FN}\cup \mathrm{NF})$ = $\frac{\text{480}}{\text{2,652}}\text{ + }\frac{\text{480}}{\text{2,652}}\text{ = }\frac{\text{960}}{\text{2,652}}\text{ = }\frac{\text{80}}{\text{221}}$
2. P ( N $|$ F ) = $\frac{40}{51}$
3. P (at most one face card) = $\frac{\text{(480 + 480 + 1,560)}}{\text{2,652}}$ = $\frac{2,520}{2,652}$
4. P (at least one face card) = $\frac{\text{(132 + 480 + 480)}}{\text{2,652}}$ = $\frac{\text{1,092}}{\text{2,652}}$

References

“Blood Types.” American Red Cross, 2013. Available online at http://www.redcrossblood.org/learn-about-blood/blood-types (accessed May 3, 2013).

Data from the National Center for Health Statistics, part of the United States Department of Health and Human Services.

Data from United States Senate. Available online at www.senate.gov (accessed May 2, 2013).

“Human Blood Types.” Unite Blood Services, 2011. Available online at http://www.unitedbloodservices.org/learnMore.aspx (accessed May 2, 2013).

Haiman, Christopher A., Daniel O. Stram, Lynn R. Wilkens, Malcom C. Pike, Laurence N. Kolonel, Brien E. Henderson, and Loīc Le Marchand. “Ethnic and Racial Differences in the Smoking-Related Risk of Lung Cancer.” The New England Journal of Medicine, 2013. Available online at http://www.nejm.org/doi/full/10.1056/NEJMoa033250 (accessed May 2, 2013).

Samuel, T. M. “Strange Facts about RH Negative Blood.” eHow Health, 2013. Available online at http://www.ehow.com/facts_5552003_strange-rh-negative-blood.html (accessed May 2, 2013).

“United States: Uniform Crime Report – State Statistics from 1960–2011.” The Disaster Center. Available online at http://www.disastercenter.com/crime/ (accessed May 2, 2013).

Data from Clara County Public H.D.

Data from the American Cancer Society.

Data from The Data and Story Library, 1996. Available online at http://lib.stat.cmu.edu/DASL/ (accessed May 2, 2013).

Data from the Federal Highway Administration, part of the United States Department of Transportation.

Data from the United States Census Bureau, part of the United States Department of Commerce.

Data from USA Today.

“Environment.” The World Bank, 2013. Available online at http://data.worldbank.org/topic/environment (accessed May 2, 2013).

“Search for Datasets.” Roper Center: Public Opinion Archives, University of Connecticut., 2013. Available online at http://www.ropercenter.uconn.edu/data_access/data/search_for_datasets.html (accessed May 2, 2013).

Chapter review

There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables help display data and are particularly useful when calculating probabilites that have multiple dependent variables.

A tree diagram use branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.