Bayes`s Theorem Practice Problems
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Bayes Theorem Practice Problems These are set of practice probability problems for exam 2. Express all probabilities to four decimal digits. Q1: Mammograms and breast cancer 1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammograms. 9.6% of women without breast cancer will also get positive mammograms. . Q1.1 What are the values of the following probabilities? ππ(πΆ) = 0.0100 Pr(πΆΜ ) = 0.9900 ππ(π|πΆ) = 0.8000 Μ |πΆ) = 0.2000 ππ(π ππ(π|πΆΜ ) = 0.0960 Μ |πΆΜ ) = 0.9400 ππ(π Bayes Theorem Practice Problems Q1.2 Draw a probability tree using the probabilities given above that can be used to compute the probability that a randomly selected woman has breast cancer if she has a positive mammogram. Bayes Theorem Practice Problems Q1.3 Draw a probability contingency table and compute the joint probabilities for each cell in the table. Event πΆ πΆΜ Marginal M 0.00800 0.09504 0.10304 Μ Marginal π 0.00200 0.0100 0.89496 0.9900 0.89696 1.0000 ππ(πΆ, π) = ππ(π|πΆ) Pr(πΆ) = 0.8000 β 0.0100 = 0.00800 Μ ) = ππ(π Μ |πΆ)ππ(πΆ) = 0.2000 β 0.0100 = 0.00200 ππ(πΆ, π ππ(πΆΜ , π) = ππ(π|πΆΜ )ππ(πΆΜ ) = 0.0960 β 0.9900 = 0.09504 Μ ) = ππ(π Μ |πΆΜ )ππ(πΆΜ ) = 0.9040 β 0.9900 = 0.89496 ππ(πΆΜ , π Q1.4 Compute the probability that a randomly chosen woman will have a positive mammogram. ππ(π) = ππ(πΆ, π) + ππ(πΆΜ , π) = 0.10304 Hint: This number can be found in the bottom row of the above table. Bayes Theorem Practice Problems Q1.5 Suppose a randomly selected woman has a mammogram and it is positive. What is the probability that she has breast cancer? What is the probability that she does not have breast cancer? ππ(πΆ|π) = ππ (πΆ,π) Pr(π) 0.00800 = 0.10304 = 0.07763975 ≈ 0.0776 Bayes Theorem Practice Problems Q2: Steroid Use in Rigby A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. Further, 15% of all steroid-free individuals also test positive. 10% of the rugby team members use steroids. Your friend on the rugby team has just tested positive. Q2.1 We define the events in this problem as: π = ππππ¦ππ π’π ππ π π‘ππππππ πΜ = ππππ¦ππ ππππ πππ‘ π’π π π π‘ππππππ π = πππ ππ‘ππ£π π‘ππ π‘ πππ π π‘ππππππ πΜ = πππππ‘ππ£π π‘ππ π‘ Given the above event definitions, what are the values of the following probabilities? ππ(π) = 0.10 ππ(πΜ ) = 0.90 ππ(π|π) = 0.95 ππ(πΜ |π) = 0.05 ππ(π|πΜ ) = 0.15 ππ(πΜ |πΜ ) = 0.85 Bayes Theorem Practice Problems Q2.2 Draw a probability tree or a contingency table that can help you compute the probability that a randomly selected rugby player is a steroid user if the player has a positive drug test. You may use the empty tree below if you wish. Draw your own table if you are using a probability contingency table. Bayes Theorem Practice Problems Q2.3 Compute the following probabilities: ππ(π, π) = ππ(π|π) ππ(π) = 0.1000 β 0.9500 = 0.0950 Pr(πΜ , π) = Pr(πΜ |π) ππ(π) = 0.0500 β 0.1000 = 0.0050 ππ(π, πΜ ) = ππ(π|πΜ )ππ(πΜ ) = 0.1500 β 0.9000 = 0.1350 ππ(πΜ , πΜ ) = ππ(π|πΜ )ππ(πΜ ) = 0.8500 β 0.9000 = 0.7650 Q2.4 Compute the probability that the steroid test will give a positive result. ππ(π) = ππ(π, π) + (πΜ , π) = 0.09050 + 0.1350 = 0.2300 Q2.4 Compute the probability that that a randomly selected player is using steroids if he has a positive test? ππ(π|π) = ππ(π, π)⁄ππ(π) = 0.0950⁄0.2300 ≈ 0.4130 Bayes Theorem Practice Problems Q2.5 A rugby team has 22 players. Suppose a league has 10 teams. How many players will be falsely implicated by this drug test if all players are tested? Let π₯ = ππ’ππππ ππ ππππ π πππ ππ‘ππ£π ππππ¦πππ . Then the expected value of x is given by: πΈ(π) = ππ = (10 β 22) β 0.1350 = 29.7 Q2.6 Do you think that this test is fair? Explain your answer. No, because approximately 30 players will be accused of using steroids when they are not steroid users.
