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Probability, SMB-014

Students studying vocational education courses were surveyed about their living arrangements.
  

  1. One of these students is selected at random. What is the probability, correct to the nearest percentage, that this student is male and living with his parent(s)?   (2 marks)

  2. A female student is selected. What is the probability that she is not living with her parent(s)?   (2 marks)

Show Answers Only
  1.  `\text{31%}`
  2. `91/114`
Show Worked Solution

i.     `text{Number of males living with parents = 155}`

`text{Total students surveyed = 505}`

`P\text{(male and living with parents)}` `=155/505`  
  `=0.3069…`  
  `=31\text{%  (nearest %)}`  

 
ii.    `text{Number of females = 228}`

`text{Females not living with parents = 182}`

`P\text{(selected female not living with parents)} = 182/228 = 91/114`

Probability, SMB-013

A group of coalminers were surveyed about what registered vehicles they own.

They were surveyed on whether they own a car, a motorbike, both or neither and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} &  &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} & 7 &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} & 29 & 6 \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} & 7 &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} & 29 & 6 \\
\hline
\end{array}

Probability, SMB-012

A group of 20 museum visitors were surveyed about what languages they could speak fluently.

They were surveyed on whether they could speak English or French and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 &  \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} & 5 & 3 \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 & 4 \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} & 5 & 3 \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 & 4 \\
\hline
\end{array}

Probability, SMB-011

A class of 30 students were surveyed about their pets. They were asked whether they owned a dog, cat, both or neither and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} &  & 8\\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} & 3 & 7 \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} & 12 & 8\\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} & 3 & 7 \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} & 12 & 8\\
\hline
\end{array}

Probability, SMB-007

Some men and women were surveyed at a football game. They were asked which team they supported. The results are shown in the two-way table.

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \textit{Team A }\ \rule[-1ex]{0pt}{0pt} &\ \textit{Team B}\ \ &\ \textit{Totals}\ \ \\
\hline
\rule{0pt}{2.5ex}\text{Men}\rule[-1ex]{0pt}{0pt} & 125 &  100 &  225 \\
\hline
\rule{0pt}{2.5ex}\text{Women}\rule[-1ex]{0pt}{0pt} & 75 & 90 & 165 \\
\hline
\rule{0pt}{2.5ex}\text{Totals}\rule[-1ex]{0pt}{0pt} & 200 & 190 & 390 \\
\hline
\end{array}

A man was chosen at random. What is the probability that he supports Team B, correct to the nearest percent?   (2 marks)

Show Answers Only

`44%`

Show Worked Solution

`text{Total number of men}\ = 225`

`text{Number of men who support Team B}\ = 100`

`P(\text{chosen man supports Team B})`

`=100/225`

`=4/9`

`=44%\ \text{(nearest %)}`

Probability, SMB-001

A group of 125 people were asked if they wear a watch or not.

This table shows the results.
 

 

A man was selected at random.

What is the exact probability that he wears a watch?   (2 marks)

Show Answers Only

`5/12` 

Show Worked Solution

`P(text(man chosen wears watch))`

`= text(number of men wearing watch)/text(total men)`

`= 25/60`

`= 5/12`

Probability, STD2 S2 2016 HSC 23 MC

A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.
 

A person is selected at random from the group.

What is the approximate probability that the person selected is a smoker OR is male?

  1. 33%
  2. 18%
  3. 68%
  4. 87%
Show Answers Only

`=> C`

Show Worked Solution

`P(text(Smoker or a male))`

`= (text(Total males + female smokers))/(text(Total surveyed))`

`= (264 + 68)/485`

`= 0.684…`
 

`=> C`

♦♦ Mean mark 34%.

Probability, STD2 S2 2006 HSC 26c

A new test has been developed for determining whether or not people are carriers of the Gaussian virus.

Two hundred people are tested. A two-way table is being used to record the results.
 

  1.  What is the value of  `A`?  (1 mark)

  2.  A person selected from the tested group is a carrier of the virus.

     

     What is the probability that the test results would show this?  (2 marks) 

  3.  For how many of the people tested were their test results inaccurate?  (1 mark)

Show Answers Only
  1. `98`
  2. `37/43`
  3. `28`
Show Worked Solution
i.  `A` `= 200-(74 + 12 + 16)`
  `= 98`

 

ii.  `P` `= text(# Positive carriers)/text(Total carriers)`
  `= 74/86`
  `= 37/43`

 

iii.  `text(# People with inaccurate results)`

`= 12 + 16`

`= 28`

Probability, STD2 S2 2005 HSC 16 MC

On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table.

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} \rule[-1ex]{0pt}{0pt} & \textbf{Male viewers} & \textbf{Female viewers} \\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 1}\rule[-1ex]{0pt}{0pt} & 1372 & 3915\\
\hline
\rule{0pt}{2.5ex}\textbf{Contestant 2}\rule[-1ex]{0pt}{0pt} & 2054 & 3269\\
\hline
\end{array}

One male viewer was selected at random from all of the male viewers.

What is the probability that he voted for Contestant 1?

  1. `1372/(10\ 610)`
  2. `1372/5287`
  3. `1372/3426`
  4. `1372/2054`
Show Answers Only

`C`

Show Worked Solution

`text(Total male viewers)\ = 1372 + 2054= 3426`

  
`P\ text{(Male viewer chosen voted for C1)}`

`= text(Males who voted for C1)/text(Total male viewers)`

`= 1372/3426`
 

`=>  C`

Probability, STD2 S2 2004 HSC 25c

Lie detector tests are not always accurate. A lie detector test was administered to 200 people.

The results were:

• 50 people lied. Of these, the test indicated that 40 had lied;
• 150 people did NOT lie. Of these, the test indicated that 20 had lied.

  1. Complete the table using the information above   (2 marks)
      
        

  2. For how many of the people tested was the lie detector test accurate?   (1 mark)

  3. For what percentage of the people tested was the test accurate?   (1 mark)

  4. What is the probability that the test indicated a lie for a person who did NOT lie?   (1 mark)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `170`
  3. `text(85%)`
  4. `2/15`
Show Worked Solution

i.

ii.  `text(# Accurate readings)`

`= 40 + 130`

`= 170`
 

iii.  `text(Percentage of people with accurate readings)`

`= text(# Accurate readings)/text(Total readings) xx 100`

`= 170/200`

`= 85 text(%)`
 

iv.  `text{P(lie detected when NOT a lie)}`

`= 20/150`

`= 2/15`

Probability, STD2 S2 2007 HSC 16 MC

Leanne copied a two-way table into her book.
 

 

Leanne made an error in copying one of the values in the shaded section of the table.

Which value has been incorrectly copied?

  1. The number of males in full-time work
  2. The number of males in part-time work
  3. The number of females in full-time work
  4. The number of females in part-time work
Show Answers Only

`D`

Show Worked Solution

`text(By checking row and column total, the number)`

`text(of females part-time work is incorrect)`

`=>  D`

Probability, STD2 S2 2008 HSC 26a

Cecil invited 175 movie critics to preview his new movie. After seeing the movie, he conducted a survey. Cecil has almost completed the two-way table.
 

  1. Determine the value of  `A`.   (1 mark)

  2. A movie critic is selected at random.

     

    What is the probability that the critic was less than 40 years old and did not like the movie?  (2 marks)

  3. Cecil believes that his movie will be a box office success if 65% of the critics who were surveyed liked the movie.

     

    Will this movie be considered a box office success? Justify your answer.  (1 mark)

Show Answers Only
  1. `58`
  2. `6/25`
  3. `text(Proof)\ \ text{(See Worked Solutions)}`
Show Worked Solution

i.  `text{Critics liked and}\ >= 40`

`= 102-65`

`= 37`

`:. A = 37+31=68`

 
ii.  `text{Critics did not like and < 40}`

`= 175-65-37-31`

`= 42`
 

`:.\ P text{(not like and  < 40)}`

`= 42/175`

`= 6/25`
 

iii.   `text(Critics liked) = 102`

`text(% Critics liked)` `= 102/175 xx 100`
  `= 58.28…%`

 
`:.\ text{Movie NOT a box office success (< 65% critics liked)}`

Probability, STD2 S2 2014 HSC 8 MC

A group of 150 people was surveyed and the results recorded.
  

A person is selected at random from the surveyed group. 

What is the probability that the person selected is a male who does not own a mobile?

  1. `28/150`
  2. `45/150` 
  3. `28/70` 
  4. `45/70` 
Show Answers Only

`A`

Show Worked Solution
`P` `= text(number of males without mobile)/text(number in group)`
  `= 28/150`

 
`=>  A`