# What Bayes theorem tell us

Mar 31, 2017

# What-Bayes theorem tell us

###If a person is infected by the disease, the probability of a positive test result will be 90%. However, if someone is not infected, there still will have 9% chances to get the positive result. We have already known that the prevalence of this disease is 1%.

##If I am tested to be positive, what is the probability of getting a true infection?

##70%?

##60%?

##50%?

##40%?

##10%?

## Answer

Just **9%**, the key is not the reliability of the test but the prevalence of this disease

the prevalence of this disease | the probability of true positive |
---|---|

1% | 9.17% |

2% | 16.95% |

3% | 23.62% |

6% | 38.96% |

9% | 49.72% |

12% | 57.69% |

15% | 63.83% |

18% | 68.7% |

21% | 72.66% |

24% | 75.95% |

27% | 78.72% |

30% | 81.08% |

33% | 83.12% |

36% | 84.91% |

39% | 86.47% |

42% | 87.87% |

45% | 89.11% |

48% | 90.23% |

51% | 91.23% |

54% | 92.15% |

57% | 92.99% |

60% | 93.75% |

63% | 94.45% |

66% | 95.1% |

69% | 95.7% |

72% | 96.26% |

75% | 96.77% |

78% | 97.26% |

81% | 97.71% |

84% | 98.13% |

87% | 98.53% |

90% | 98.9% |

93% | 99.25% |

96% | 99.59% |

99% | 99.9% |

the probability of a positive test result if infected | the probability of a positive test result if not infected |
---|---|

90% | 3% |

the prevalence of this disease | the probability of true positive |
---|---|

1% | 23.26% |

2% | 37.97% |

3% | 48.13% |

6% | 65.69% |

9% | 74.79% |

12% | 80.36% |

15% | 84.11% |

18% | 86.82% |

21% | 88.86% |

24% | 90.45% |

27% | 91.73% |

30% | 92.78% |

33% | 93.66% |

36% | 94.41% |

39% | 95.04% |

42% | 95.6% |

45% | 96.09% |

48% | 96.51% |

51% | 96.9% |

54% | 97.24% |

57% | 97.55% |

60% | 97.83% |

63% | 98.08% |

66% | 98.31% |

69% | 98.52% |

72% | 98.72% |

75% | 98.9% |

78% | 99.07% |

81% | 99.22% |

84% | 99.37% |

87% | 99.5% |

90% | 99.63% |

93% | 99.75% |

96% | 99.86% |

99% | 99.97% |

the probability of a positive test result if infected | the probability of a positive test result if not infected |
---|---|

80% | 3% |

the prevalence of this disease | the probability of true positive |
---|---|

1% | 23.26% |

2% | 37.97% |

3% | 45.2% |

6% | 62.99% |

9% | 72.51% |

12% | 78.43% |

15% | 82.47% |

18% | 85.41% |

21% | 87.64% |

24% | 89.39% |

27% | 90.79% |

30% | 91.95% |

33% | 92.93% |

36% | 93.75% |

39% | 94.46% |

42% | 95.08% |

45% | 95.62% |

48% | 96.1% |

51% | 96.52% |

54% | 96.9% |

57% | 97.25% |

60% | 97.56% |

63% | 97.85% |

66% | 98.1% |

69% | 98.34% |

72% | 98.56% |

75% | 98.77% |

78% | 98.95% |

81% | 99.13% |

84% | 99.29% |

87% | 99.44% |

90% | 99.59% |

93% | 99.72% |

96% | 99.84% |

99% | 99.96% |