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Independence and Base Rates worksheets

Check independence in rational tables and compare Bayesian posteriors under different priors.

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Use joint and marginal probabilities to check independence. In Bayes problems, keep the prior visible.

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Learning objective

What this activity builds

Distinguish independence from association and preserve the prior when reversing conditional probabilities.

Worked example

If P(A)=1/2, P(B)=1/4, and P(A and B)=1/8, are A and B independent?

Yes, because (1/2)(1/4)=1/8.

The joint equals the product of marginals.

Teaching notes

Make the practice count

  • Complete all four table cells.
  • Compare posteriors with the same likelihoods.
  • Name the direction of every conditional.

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Math SheetNameScoreTeacherDateIndependence and Base RatesUse joint and marginal probabilities to check independence. In Bayes problems, keep the prior visible.1.The same test has sensitivity 0.9 and false-positive rate 0.2. Compare P(condition | positive) in populations with priors 0.1 and 0.3.Reference: Bayes: posterior=(prior x sensitivity)/[(prior x sensitivity)+(1-prior)x false-positive rate].2.A condition has prior 0.1. A test has P(positive | condition)=0.75 and P(positive | no condition)=0.1. A student claims P(condition | positive)=0.75. Explain theerror and find the correct posterior.Reference: Reversing a conditional changes its denominator. Use the prior and total positive probability.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateIndependence and Base RatesUse joint and marginal probabilities to check independence. In Bayes problems, keep the prior visible.3.A 96-person table has 48 in A, 24 in B, and 12 in both A and B. Complete the four cell counts and decide whether A and B are independent.Reference: Events are independent exactly when P(A intersection B)=P(A)P(B).4.The same test has sensitivity 0.8 and false-positive rate 0.2. Compare P(condition | positive) in populations with priors 0.1 and 0.3.Reference: Bayes: posterior=(prior x sensitivity)/[(prior x sensitivity)+(1-prior)x false-positive rate].State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateIndependence and Base RatesUse joint and marginal probabilities to check independence. In Bayes problems, keep the prior visible.5.A 112-person table has 56 in A, 28 in B, and 14 in both A and B. Complete the four cell counts and decide whether A and B are independent.Reference: Events are independent exactly when P(A intersection B)=P(A)P(B).6.A condition has prior 0.1. A test has P(positive | condition)=0.75 and P(positive | no condition)=0.05. A student claims P(condition | positive)=0.75. Explain theerror and find the correct posterior.Reference: Reversing a conditional changes its denominator. Use the prior and total positive probability.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateIndependence and Base RatesUse joint and marginal probabilities to check independence. In Bayes problems, keep the prior visible.7.A 128-person table has 64 in A, 32 in B, and 16 in both A and B. Complete the four cell counts and decide whether A and B are independent.Reference: Events are independent exactly when P(A intersection B)=P(A)P(B).8.The same test has sensitivity 0.8 and false-positive rate 0.1. Compare P(condition | positive) in populations with priors 0.01 and 0.4.Reference: Bayes: posterior=(prior x sensitivity)/[(prior x sensitivity)+(1-prior)x false-positive rate].State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 4 of 4

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