# 6.5. Independent Events

# Independent Events

In the previous section, we considered conditional probabilities. In some examples, the probability of an event changed when additional information was provided. For instance, the probability of obtaining a king from a deck of cards changed from 4/52 to 4/12 when we were given the condition that a face card had already shown. This is not always the case. The additional information may or may not alter the probability of the event. For example consider the following example.

Example 6.5.1

**a.**The card is a king.

**b.**The card is a king given that a red card has shown.

**Solution**

**a.**Clearly,

*P*(The card is a king) = 4/52 = 1/13.

**b.**To find

*P*(The card is a king | A red card has shown), we reason as follows:

*P*(The card is a king | A red card has shown) = 2/26 = 1/13.

The reader should observe that in the above example:

*P*(The card is a king | A red card has shown) =* P*(The card is a king)

In other words, the additional information, a red card has shown, did not affect the probability of obtaining a king. Whenever the probability of an event *E* is not affected by the occurrence of another event *F*, and vice versa, we say that the two events *E* and *F* are **independent**. This leads to the following definition.

Two events *E* and *F* are **independent** if and only if at least one of the following two conditions is true:

1.

or

2.

If the events are not independent, then they are dependent. We can test for independence with the following formula.

**Test for Independence**

Two Events *E* and *F* are independent if and only if

Example 6.5.2

Male (M) | Female (F) | Total | |

Colour-Blind (C) | 6 | 1 | 7 |

Not Colour-Blind (N) | 46 | 47 | 93 |

Total | 52 | 48 | 100 |

*M*represents male,

*F*represents female,

*C*represents colour-blind, and

*N*not colour-blind. Use the independence test to determine whether the events colour-blind and male are independent.

**Solution**

*C*and

*M*are independent if and only if .

Example 6.5.3

**Solution**

*H*be the event that an adult owns a home, and

*D*the event that an adult had diabetes. We have:

Example 6.5.4

*E*,

*F*and

*G*are defined as follows:

**a.**E and F

**b**. F and G

**c.**E and G

**Solution**

*S*= {HHH , HHT , HTH , HTT , THH , THT , TTH , TTT}

*E*= {HHH , HHT , HTH , HTT},

*F*= {HHH , HHT , HTH , THH},

*G*= {HHT , THH},

= {HHT},

**a.**In order for

*E*and

*F*to be independent, we must have:

*E*and

*F*are not independent.

**b.**

*F*and

*G*will be independent if:

*F*and

*G*are not independent.

**c.**We look at :

*E*and

*G*are independent events.

Example 6.5.5

**Solution**

*A*be the event that Jaime will visit his aunt this year, and R be the event that he will go river rafting.

*P*(

*A*) = 0.30 and

*P*(

*R*) = 0.50, and we want to find .

*A*and

*R*are independent:

Example 6.5.6

*P*(

*B*|

*A*) = 0.4. If

*A*and

*B*are independent, find

*P*(

*B*).

**Solution**

*A*and

*B*are independent, then by definition

*P*(

*B*|

*A*) =

*P*(

*B*).

*P*(

*B*) = 0.4

Example 6.5.7

*P*(

*A*) = 0.7,

*P*(

*B*|

*A*) = 0.5. Find .

**Solution**

Example 6.5.8

*A*and

*B*are independent, find .

**Solution**

*A*and

*B*are independent,

# Practice questions

**1.** In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 10 had minor headaches. Of the non-drinkers, 5 had minor headaches. Are the events “drinkers” and “had headaches” independent?

**2.** Suppose that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events wearing a seat belt and quitting smoking independent?

**3.** If , , and E and F are independent, find .

**4.** John’s probability of passing Data Management is 40%, and Linda’s probability of passing the same course is 70%. If the two events are independent, find the following probabilities:

**a. ***P *(both of them will pass the course)

**b.** *P *(at least one of them will pass the course)

**5.** The table below shows the distribution of employees in a company that reported a previous workplace injury based on their years of working experience at the company.

Less than 10 years of experience (L) | 10 or more years of experience (E) | Total | |

Did not report a workplace injury (N) | 300 | 100 | 400 |

Reported a workplace injury (Y) | 150 | 50 | 200 |

450 | 150 | 600 |

**a. **

**b.**

**c.**

**d.** Are the events L and Y independent?

**6. **Given , , if *A* and *B* are independent, find .