# Statistics for Decision Makers - 05.05 - Probability - Base Rate Fallacy

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## Drunk Driving。

- Breathalysers display a false result in 5% of the cases tested
- They never fail to detect a truly drunk person
- 1/1000 of drivers are driving drunk
- Policemen then stop a driver
**at random**, and test them - The breathalyser indicates that the driver is drunk

- How high is the probability the driver is really drunk?

Result\Reality | Drunk | Sober |
---|---|---|

Test Positive | 1 | 0.05 |

Test Negative | 0 | 0.95 |

Historic Data | 0.001 | 0.999 |

### Drunk Driving。

Let us assume we tested 1,000,000 people.

True Condition | |||
---|---|---|---|

Drunk | Not Drunk | ||

1,000 | 999,000 | ||

Positive | Negative | Positive | Negative |

1,000 | 0 | 49,950 | 949,050 |

- How many of those people tested positive?
- How many of those who tested positive were really drunk?
- What is a "favourable" outcome?
- How many "favourable" outcomes are there?
- How many possible outcomes are there?
- What is the probability of a person who is tested being really drunk?

### Drunk Driving。

Let us assume we tested 1,000,000 people.

True Condition | |||
---|---|---|---|

Drunk | Not Drunk | ||

1,000 | 999,000 | ||

Positive | Negative | Positive | Negative |

1,000 | 0 | 49,950 | 949,050 |

- How many of those people tested positive?

1,000 + 49,950 = 50,950

- How many of those who tested positive were really drunk?

1000

- What is the probability of a person who is tested being really drunk?

1000/50,950=0.01962

### Drunk Driving。

## Base Rate Fallacy。

- The Base Rate in our case is 0.001 and 0.999 probabilities.
- An overwhelming proportion of people are sober, therefore the probability of a false positive (5%) is much more prominent than the 100% probability of a true positive.
- People tend to
**simply ignore**the base rates, hence it is called (**base rate neglect**). - In other words, no matter what the base rates, people tend to look at only the "test accuracy rate".

## Base Rate Fallacy Examples。

- Detecting terrorists
- Detecting a rare disease
- Detecting prospective customers (provided that most people will not buy our product)
- Some DNA tests

## Bayes theorem。

- What is the probability that a driver is drunk given that the breathalyser indicates that he/she is drunk?

Bayes' Theorem tells us that:

We were told the following in the first paragraph:

After using Bayes' Theorem:

## Is a promotion really working?。

An online advertising company knows (based on its historical record) that 10% of the people who try the trial version of their services will convert into paying customers.

- You propose to introduce a promotion: each new customer will be granted a free $100 for advertising
- You know that some people will just register to use the $100 even if they do not intend to convert into paying customers
- You want to test the effectiveness of the free $100 promotion
- After running the promotion, 40% of customers who converted used the $100 promotion
- Also, 10% of prospects who did not convert used the promotion

### Is the promotion really working?。

- Does the promotion increase the probability of conversion?

- Events

Prom: Customer uses the promotion NotProm: Customer does not use the promotion Con: Customer converts NotCon: Customer does not convert

- P(Con) = 0.1
- An online advertising company knows, based on its historical record, that 10% of the people who try the trial version of their services will convert into customers
- P(Prom|Con) = 0.4
- After running the promotion, 40% of customers who converted used the $100 promotion
- P(Prom|NotCon) = 0.1
- Also, 10% of prospects who used the promotion did not convert

- Compute complementary probabilities

P(NotCon) = 0.9 P(NotProm|Con) = 0.6 P(NotProm|NotCon) = 0.9

- Does the promotion increase probability of conversion?

P(Con|Prom) > P(Con) = 0.1 P(Con|Prom) from BaysTheorm = 0.308

# Quiz。

# Quiz