Lesson 3.4 Contingency Tables

Contingency Tables

A contingency table is a tool that simplifies calculating probabilities, especially conditional probabilities. The table displays sample values in relation to two different variables that may be contingent (dependent) on one another.

Example: All current carrying wires produce electromagnetic radiation. High frequency radiation is sometimes thought to be the cause of cancer, while low frequency radiation is generally assumed to be harmless. The table below presents the results of a study in a large U.S. metropolitan area of 215 children who died of cancer and the frequency radiation produced by the wires outside the building where the children lived.

---	Leukemia	Lymphoma	Other Cancers	RowTotal
High Frequency	52	10	17	79
Low Frequency	84	21	31	136
Column Total	136	31	48	215

What is the probability that a randomly chosen child died of lymphoma?

The solution is P(Lymphoma) = 31/215.

What is the probability that one of the randomly chosen children died from high frequency radiation exposure?

The solution is P(High Frequency) = 79/215.

What is the probability that a randomly chosen child died of other cancers and had low frequency radiation exposure?

There are 31 children who died of other cancers and had low frequency exposure. The solution is:

P(Other Cancers AND Low Frequency) = 31/215.

What is the probability that a randomly chosen child died of other cancers or had low frequency radiation exposure? To clearly understand the solution, use the Addition Rule.

P(Other Cancers OR Low Frequency)

= P(Other Cancers) + P(Low Frequency) - P(Other Cancers AND Low Frequency)

= 48/215 + 136/215 - 31/215 = 153/215

What is the probability that a randomly chosen child died of leukemia GIVEN that the child had high frequency radiation exposure?

This is a conditional probability. The sample space has been reduced to the children who had high frequency radiation exposure (79 children). Of those children who had high frequency radiation exposure, 52 died of leukemia.

P(Leukemia | High Frequency Radiation) = 52/79.

What is the probability that a randomly chosen child had high frequency radiation exposure GIVEN that the child died of leukemia? Again, this is a conditional probability. The sample space has been reduced to the 136 children who died of leukemia.

P(High Frequency|Leukemia) = 52/136.

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Contingency Table Problem

Example

The following example is a contingency table probability problem concerning hair color and hair type. Close the window when you are finished viewing the example. You will return here.

This is the last section of this lesson.

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Up » 3.1 Terminology » 3.2 Independent or Mutually Exclusive » 3.3 Addition and Multiplication Rules » 3.4 Contingency Tables