Problem preview

S-MD.7 Core M3-055-R05-V02

Analyze decisions and strategies using probability concepts in more complex settings.

Update a state probability after new information

Problem

Fraud prior is \(0.02\), \(P(\text{flag}~|~\text{fraud})~=~0.90\), and \(P(\text{flag}~|~\text{not}~\text{fraud})~=~0.05\). An item is flagged. Compute both flag joint branches, \(P(\text{flag})\), \(P(\text{fraud}~|~\text{flag})\), and compare posterior with prior.

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Keep the prior .02 as posterior despite flag branches .018 and .049 and total .067; report no update.

Use sensitivity .90 as the posterior; joint branches are .018 and .049, P(flag)=.067, so incorrectly report .90.

P(fraud and flag)=.018 and P(flag)=.067, so P(fraud|flag)=18/67 ≈ .269, an increase from .02.

Use the nonfraud flag branch .049/.067=.731 as P(fraud|flag), producing the complement of the correct posterior.

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Curriculum context

Course: Math III
Standard: S-MD.7
Category: Statistics and Probability
Domain: Using Probability to Make Decisions
Objective: Analyze decisions and strategies using probability concepts in more complex settings.
Problem type: Update a state probability after new information