# 7.2. Bayes’ Formula

# Bayes’ Formula

In this section, we will develop and use Bayes’ Formula to solve an important type of probability problem. Bayes’ formula is a method of calculating the conditional probability *P*(*F* | *E*) from *P*(*E* | *F*). The ideas involved here are not new, and most of these problems can be solved using a tree diagram. However, Bayes’ formula does provide us with a tool with which we can solve these problems without a tree diagram. We begin with an example.

Example 7.2.1

**a.**What is the probability that the marble chosen is a black marble?

**b.**If the chosen marble is black, what is the probability that it came from Jar I?

**c.**If the chosen marble is black, what is the probability that it came from Jar II?

**Solution**

Let *JI* be the event that Jar I is chosen, *JII* be the event that Jar II is chosen, *B* be the event that a black marble is chosen and *W* the event that a white marble is chosen. We illustrate using a tree diagram.

(a) |
(b) |

**a.**The probability that a black marble is chosen is

*P*(

*B*) = 1/10 + 2/10 = 3/10.

**b.**To find

*P*(

*JI*|

*B*), we use the definition of conditional probability, and we get

**c.**Similarly,

**b**and

**c**, the reader should note that the denominator is the sum of all probabilities of all branches of the tree that produce a black marble, while the numerator is the branch that is associated with the particular jar in question.

This is a statement of Bayes’ formula.

**Bayes’ Formula:**Let *S* be a sample space that is divided into *n* partitions, *A*_{1}, *A*_{2}, . . . *A _{n}*. If

*E*is any event in

*S*, then:

Example 7.2.2

**Solution**

*A*,

*B*and

*C*be the events that the appliance is manufactured by Manufacturer A, Manufacturer B, and Manufacturer C, respectively. Further, suppose that the event

*R*denotes that the appliance needs repair before the warranty expires.

*P*(

*A*|

*R*),

*P*(

*B*|

*R*) and

*P*(

*C*|

*R*).

The probability *P*(*A* | *R*), for example, is a fraction whose denominator is the sum of all probabilities of all branches of the tree that result in an appliance that needs repair before the warranty expires, and the numerator is the branch that is associated with Manufacturer A. *P*(*B* | *R*) and *P*(*C* | *R*) are found in the same way. We list both as follows:

*P*(

*B*|

*R*) and

*P*(

*C*|

*R*) can be determined in the same manner.

Example 7.2.3

Store Number | Number of Employees | Proportion of Women Employees |

1 | 300 | 0.40 |

2 | 150 | 0.65 |

3 | 200 | 0.60 |

4 | 250 | 0.50 |

5 | 100 | 0.70 |

Total = 1000 |

**Solution**

*k*= 1, 2, …, 5 be the event that the employee worked at store

*k*, and

*W*be the event that the employee is a woman. Since there are a total of 1000 employees at the five stores,

*P*(1) = 0.30

*P*(2) = 0.15

*P*(3) = 0.20

*P*(4) = 0.25

*P*(5) = 0.10

For certain problems, we can use a much more intuitive approach than Bayes’ Formula.

Example 7.2.4

**false negative rate**is 10% (that is, about 10% of people who take the test will test negative, even though they actually have the disease). The

**false positive rate**is 1% (that is, about 1% of people who take the test will test positive, even though they do not actually have the disease). Compute the probability that a person who tests positive actually has the disease:

Positive test | Negative test | Total | |

Have disease | 180 | 20 | 200 |

Do not have disease | 98 | 9,702 | 9,800 |

Total | 278 | 9,822 | 10,000 |

**Solution**

# Practice questions

**1.** Jar I contains five red and three white marbles, and Jar II contains four red and two white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram and find the following probabilities:

**a. ***P *(Marble is red)

**b. **P (The marble came from Jar II given that a white marble is drawn)

**c.** *P *(Red marble | Jar I)

**2.** The table below summarizes the results of a diagnostic test:

Positive test | Negative test | Total | |

Have disease | 105 | 15 | 120 |

Do not have disease | 40 | 640 | 680 |

Total | 145 | 655 | 800 |

Using the table, compute the following:

**a.** P (Negative test | disease positive)

**b.** P (Disease positive | test positive)

**3****.** A computer company buys its chips from three different manufacturers. Manufacturer I provides 60% of the chips, of which 5% are known to be defective; Manufacturer II supplies 30% of the chips, of which 4% are defective; while the rest are supplied by Manufacturer III, of which 3% are defective. If a chip is chosen at random, find the following probabilities:

**a. ***P *(The chip is defective)

**b. **P (The chip came from Manufacturer II | it is defective)

**c.** *P *(The chip is defective | it came from manufacturer III)

**4. **The following table shows the percent of “Conditional Passes” that different types of food premises received in a city during their last public health inspection.

Premise Type | Number of Premises | Proportion that Received Conditional Pass |

Restaurant | 2000 | 0.07 |

Grocery Store | 425 | 0.03 |

Cafe/Bar | 1865 | 0.05 |

Food Truck/Cart | 150 | 0.08 |

Other | 560 | 0.05 |

Total = 5000 |

**a. ***P *(Received Conditional Pass)

**b. ***P *(Received Conditional Pass | Restaurant)

**c. ***P *(Grocery Store | Received Conditional Pass)