Chapter 9 : Probability of Combined Events

Chapter 9 : Probability of Combined Events

• Combined events
• Dependent events and independent events
• Mutually exclusive events and non-mutually exclusive events

Combined Event

• 3 balls in a basket labelled 1, 2 and 3

• one ball taken from the basket and returned

• second ball chosen randomly

• Sample space, S

•  $S\ =\ \left\{\left(1,1\right),\ \left(1,\ 2\right),\ \left(1,\ 3\right),\ \left(2,1\right),\left(2,2\right),\left(2,3\right),\ \left(3,1\right),\ \left(3,2\right),\ \left(3,3\right)\right\}$  

Venn Diagram for previous question (combined event)

keywords:

•  $OR\ \rightarrow\cup$  
  

•  $AND\ \rightarrow\cap$  

• Venn diagram

•  $universal\ set\ ,\ \xi$  

•  $P\left(A\right)$  : Probability of event A

Dependent Events and independent events

A) choose number 2 on the second choice

B) choose number 2 or 3 on the first choice

C) choose number 2 on the first choice and number 1 on the second choice

A) choose number 2 on second choice

• Let A = the event chose number 2 on second choice

•  $=\left\{\left(1,2\right),\ \left(2,2\right),\ \left(3,2\right)\right\}$  

•  $=P\left(number\ 2\right)$  

•  $P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{3}{9}=\frac{1}{3}$  

B) choose number 2 or 3 on first choice

• Let B = the event chose number 2 on the first choice

• Let C = the event chose number 3 on the first choice

• Question :  $B\ atau\ C\ =\ B\ \cup C$  

B) choose number 2 or 3 on first choice

•  $B=\ \left\{\left(2,1\right),\ \left(2,2\right),\ \left(2,3\right)\right\}$  
  

•  $C=\left\{\left(3,1\right),\ \left(3,2\right),\ \left(3,3\right)\right\}$  
  

• Question :  $B\ atau\ C\ =\ B\ \cup C$  

•  $\left\{\left(2,1\right),\ \left(2,2\right),\ \left(2,3\right),\ \left(3,1\right),\left(3,2\right),\ \left(3,3\right)\right\}$  

•  $n\left(B\cup C\right)=6$  

B) choose number 2 or 3 on first choice

•  $\therefore P\left(number\ 2\ or\ number\ 3\right)$  
  

•  $=P\left(B\cup C\right)$  

•  $\frac{n\left(B\cup C\right)}{n\left(S\right)}$  

•  $\frac{6}{9}=\frac{2}{3}$  

Alternative Method:

•  $B=\ \left\{\left(2,1\right),\ \left(2,2\right),\ \left(2,3\right)\right\}$  
  

•  $C=\left\{\left(3,1\right),\ \left(3,2\right),\ \left(3,3\right)\right\}$  
  

•  $=P\left(B\cup C\right)$  
  

•  $=P\left(B\right)+P\left(C\right)$  

•  $\frac{1}{3}+\frac{1}{3}$  

•  $\frac{2}{3}$  

C) choose number 2 on the first choice and number 1 on the second choice

• Let B = choose number 2 on the first choice

• Let D = choose number 1 on the second choice

• Question :  $B\ and\ D\ =\ B\ \cap\ D$  

C) choose number 2 on the first choice and number 1 on the second choice

• B =  $\left\{\left(2,1\right),\ \left(2,2\right),\ \left(2,3\right)\right\}$  

• D =  $\left\{\left(1,1\right),\left(2,1\right),\left(3,1\right)\right\}$  

• Question :  $B\ and\ D\ =\ B\ \cap\ D$  

•  $=\left\{\left(2,1\right)\right\}$  

•  $n\left(B\cap D\right)=1$  

C) choose number 2 on the first choice and number 1 on the second choice

•  $\therefore\ P\left(number\ 2\ and\ number\ 1\right)$  

•  $P\left(B\cap D\right)=\frac{n\left(B\cap D\right)}{n\left(s\right)}$  

•  $=\frac{1}{9}$  

Alternative Method:

•  $P\left(number\ 2\ AND\ number\ 1\right)$  

•  $=P\left(B\cap D\right)$  

•  $=\frac{1}{9}$  

• (refer to the Venn Diagram)

AND  $\rightarrow\ \cap$  

• First chosen MUST number 2 

•  $AND$  

• Second chosen MUST number 1

Poll

DO YOU UNDERSTAND THE CONCEPT OF COMBINED EVENT?

• YES

• NO

Poll

DO YOU UNDERSTAND THE CONCEPT OF DEPENDENT EVENT AND INDEPENDENT EVENT?

• YES

• NO