# 0.11 Independent and dependent events

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## Introduction

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

## 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.

Independent Events

Two events $A$ and $B$ are independent if when one of them happens, it doesn't affect whether the one happens or not.

The probability of two independent events occurring, $P\left(A\cap B\right)$ , is given by:

$P\left(A\cap B\right)=P\left(A\right)×P\left(B\right)$

What is the probability of rolling a 1 and then rolling a 6 on a fair die?

1. 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.

2. The probability of rolling a 1 is $\frac{1}{6}$ and the probability of rolling a 6 is $\frac{1}{6}$ .

Therefore, $P\left(A\right)=\frac{1}{6}$ and $P\left(B\right)=\frac{1}{6}$ .

3. $\begin{array}{ccc}\hfill P\left(A\cap B\right)& =& P\left(A\right)×P\left(B\right)\hfill \\ & =& \frac{1}{6}×\frac{1}{6}\hfill \\ & =& \frac{1}{36}\hfill \end{array}$

The probability of rolling a 1 and then rolling a 6 on a fair die is $\frac{1}{36}$ .

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

A cloth bag has four coins, one R1 coin, two R2 coins and one R5 coin. What is the probability of first selecting a R1 coin followed by selecting a R2 coin?

1. 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.

2. The probability of first selecting a R1 coin is $\frac{1}{4}$ and the probability of next selecting a R2 coin is $\frac{2}{3}$ (because after the R1 coin has been selected, there are only three coins to choose from).

Therefore, $P\left(A\right)=\frac{1}{4}$ and $P\left(B\right)=\frac{2}{3}$ .

3. The same equation as for independent events are used, but the probabilities are calculated differently.

$\begin{array}{ccc}\hfill P\left(A\cap B\right)& =& P\left(A\right)×P\left(B\right)\hfill \\ & =& \frac{1}{4}×\frac{2}{3}\hfill \\ & =& \frac{2}{12}\hfill \\ & =& \frac{1}{6}\hfill \end{array}$

The probability of first selecting a R1 coin followed by selecting a R2 coin is $\frac{1}{6}$ .

## 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.

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.

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 success independent of gender? Explain.
2. Give a table for the independence of gender results.
1.  Male Female Totals Positive result 50 40 90 No Positive result 70 50 120 Totals 120 90 210
2. P(male).P(positive result)= $\frac{120}{210}=0,57$

P(female).P(positive result)= $\frac{90}{210}=0,43$

P(male and positive result)= $\frac{50}{210}=0,24$

3. 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 medication's success is independent of gender.

4. To get gender independence we need the positive 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 $\frac{4}{7}$ of the total number of patients responding positively which gives 51,4. This leads to the following table:

 Male Female Totals Positive result 51,4 38,6 90 No Positive result 68,6 51,4 120 Totals 120 90 210

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