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INDEPENDENT AND DEPENDENT EVENTS - GRADE 11

34.1 Introduction

In probability theory an event is either independent or dependent. This chapter describes the
differences and how each type of event is worked with.

34.2 Definitions

Two events are independent if knowing something about the value of one event does not give
any information about the value of the second event. For example, the event of getting a ”1”
when a die is rolled and the event of getting a ”1” the second time it is thrown are independent.

Definition: Independent Events
Two events A and B are independent if when one of them happens, it doesn’t affect the
other one happening or not.

The probability of two independent events occurring, P(A ∩ B), is given by:

Worked Example 151: Independent Events

Question: What is the probability of rolling a 1 and then rolling a 6 on a fair die?
Answer
Step 1 : Identify the two events and determine whether the events are
independent or not

Event A is rolling a 1 and event B is rolling a 6. Since the outcome of the first
event does not affect the outcome of the second event, the events are independent.
Step 2 : Determine the probability of the specific outcomes occurring, for
each event

The probability of rolling a 1 is   and the probability of rolling a 6 is .
Therefore, P(A) = and P(B) = .
Step 3 : Use equation 34.1 to determine the probability of the two events
occurring together.

The probability of rolling a 1 and then rolling a 6 on a fair die is .

 

Consequently, two events are dependent if the outcome of the first event affects the outcome of
the second event.

Worked Example 152: Dependent Events

Question: A cloth bag has 4 coins, 1 R1 coin, 2 R2 coins and 1 R5 coin. What is
the probability of first selecting a R1 coin followed by selecting a R2 coin?
Answer
Step 1 : Identify the two events and determine whether the events are
independent or not

Event A is selecting a R1 coin and event B is next selecting a R2. Since the outcome
of the first event affects the outcome of the second event (because there are less
coins to choose from after the first coin has been selected), the events are dependent.
Step 2 : Determine the probability of the specific outcomes occurring, for
each event

The probability of first selecting a R1 coin is and the probability of next selecting
a R2 coin is (because after the R1 coin has been selected, there are only three
coins to choose from).
Therefore, P(A) = and P(B) = .
Step 3 : Use equation 34.1 to determine the probability of the two events
occurring together.

The same equation as for independent events are used, but the probabilities are


calculated differently .

The probability of first selecting a R1 coin followed by selecting a R2 coin is .

 

34.2.1 Identification of Independent and Dependent Events

Use of a Contingency Table


A two-way contingency table (studied in an earlier grade) can be used to determine whether
events are independent or dependent.

Definition: two-way contingency table
A two-way contingency table is used to represent possible outcomes when two events are
combined in a statistical analysis.

For example we can draw and analyse a two-way contingency table to solve the following problem.

Worked Example 153: Contingency Tables

Question: A medical trial into the effectiveness of a new medication was carried
out. 120 males and 90 females responded. Out of these 50 males and 40 females
responded positively to the medication .
1. Was the medication’s succes independent of gender? Explain.
2. Give a table for the independent of gender results.
Answer
Step 1 : Draw a contingency table
 
  Male Female Totals
Positive result
No Positive result
50
70
40
50
90
120
Totals 120 90 210

Step 2 : Work out probabilities
P(male).P(positive result)= = 0.57
P(female).P(positive result)= = 0.43
P(male and positive result)= = 0.24
Step 3 : Draw conclusion
P(male and positive result) is the observed probability and P(male).P(positive result)
is the expected probability. These two are quite different . So there is no evidence
that the medications success is independent of gender.
Step 4 : Gender-independent results
To get gender independence we need the positve results in the same ratio as the
gender
. The gender ratio is: 120:90, or 4:3, so the number in the male and positive
column would have to be of the total number of patients responding positively
which gives 22. This leads to the following table:

  Male Female Totals
Positive result
No Positive result
22
98
68
22
90
120
Totals 120 90 210
 
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