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Lecture 9

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STAT 101Professor

Richard WatermanLecture

9This

**preview**shows page 1. to view the full**5 pages of the document.**Stat 101 - Introduction to Business Statistics - Lecture 9: Bayes Theorem

Cancer Example

● You have a cancer screening test. It comes back positive. How much should you worry?

You want to find the probability of cancer given a positive screening test

○ **** Ultimately we wish to find P(C|D).

● Required facts:

○ The probability a test comes back negative given no cancer (specificity).

■ 1 - false positive rate

○ The probability a test comes back positive given cancer (sensitivity).

■ 1 - false negative rate

○ The probability of cancer (with no conditioning). (marginal probability)

● Define variables

○ Call C the event that cancer is present.

○ Call Cc the event cancer is not present.

○ Call D the event the diagnostic test comes back positive.

○ call Dc the event the diagnostic test comes back negative.

● Available information:

○ Specificity = P(Dc |Cc )=0.925.

○ Sensitivity = P(D|C)=0.825.

○ Population prevalence = P(C)=0.005.

○ **Note: The key insight is that these conditional probabilities will allow us to

propagate the marginal probabilities back into the cells of the contingency table.

● Now, construct a contingency table

○ Step 1: Work out cell counts (assume here that 1000 people have test done)

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○ Step 2: fill in the margin based on the prevalence of cancer

○ Step 3: Use the conditional probabilities to propagate the marginal probabilities to

the cells. Remember “joint equals marginal times conditional”

○ Step 4: Work out the numbers, including second margin

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