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Probability, STD2 EQ-Bank 5 MC

The Venn diagram shows information about 40 people surveyed about whether they own a dog \((D)\) or a cat \((C)\).

Which two-way table correctly represents the information in the Venn diagram?

A.      \(\begin{array}{|l|c|c|c|} \hline & D & \text{Not }D & \text{Total} \\ \hline C & 12 & 8 & 20 \\ \hline \text{Not }C & 5 & 15 & 20 \\ \hline \text{Total} & 17 & 23 & 40 \\ \hline \end{array}\) B.    \(\begin{array}{|l|c|c|c|} \hline & D & \text{Not }D & \text{Total} \\ \hline C & 8 & 12 & 20 \\ \hline \text{Not }C & 15 & 5 & 20 \\ \hline \text{Total} & 23 & 17 & 40 \\ \hline \end{array}\)
C.    \(\begin{array}{|l|c|c|c|} \hline & D & \text{Not }D & \text{Total} \\ \hline C & 15 & 12 & 27 \\ \hline \text{Not }C & 8 & 5 & 13 \\ \hline \text{Total} & 23 & 17 & 40 \\ \hline \end{array}\) D.    \(\begin{array}{|l|c|c|c|} \hline & D & \text{Not }D & \text{Total} \\ \hline C & 5 & 12 & 17 \\ \hline \text{Not }C & 15 & 8 & 23 \\ \hline \text{Total} & 20 & 20 & 40 \\ \hline \end{array}\)
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Reading the Venn diagram:}\)

\(\text{C and D (intersection)} = 8\)

\(\text{C only (not D)} = 12\)

\(\text{D only (not C)} = 15\)

\(\text{Neither} = 5\)
  

\(\text{In the two-way table:}\)

\(\text{Row }C\text{: }\ D\text{ column} = 8,\ \text{Not }D\text{ column} = 12,\ \text{Total} = 20\)

\(\text{Row Not }C\text{: }\ D\text{ column} = 15,\ \text{Not }D\text{ column} = 5,\ \text{Total} = 20\)
  

\(\Rightarrow B\)

Probability, STD2 S2 2024 HSC 12 MC

A survey of 370 people was conducted to investigate the association between watching Anime and the age of the person.

The two-way table shows the responses collected.

Approximately what percentage of the over 30-year-olds watch Anime?

  1. 9%
  2. 18%
  3. 22%
  4. 42%
Show Answers Only

\(C\)

Show Worked Solution

\(\text {Total over } 30=157\)

\(\text {Over 30s who watch anime = 34}\)

\(\text {% over 30s who watch anime}\ =\dfrac{34}{157}\times 100=21.7\%\)

\(\Rightarrow C\)

Probability, STD2 S2 2017 HSC 29c

A group of Year 12 students was surveyed. The students were asked whether they live in the city or the country. They were also asked if they have ever waterskied.

The results are recorded in the table.
  

  1. A person is selected at random from the group surveyed. Calculate the probability that the person lives in the city and has never waterskied.   (2 marks)

  2. A newspaper article claimed that Year 12 students who live in the country are more likely to have waterskied than those who live in the city.

     

    Is this true, based on the survey results? Justify your answer with relevant calculations.   (2 marks)

Show Answers Only

i.    `125/176`

ii.   `text(See Worked Solutions)`

Show Worked Solution

i.    `P= text(live in city, not skied)/text(total surveyed)= 2500/3520= 125/176`

♦ Mean mark part (ii) 47%.

 
ii.   `P(text(live in country, skied))= 70/((70 + 800))= 0.0804…= 8text(%)`

`P(text(live in city, skied))= 150/((150 + 2500))= 0.0566= 6text(%)`

`text(S)text(ince 8% > 6%, the statement is true.)`

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.
  3. What is the probability that the test results would show this?   (2 marks) 
  4.  For how many of the people tested were their test results inaccurate?   (1 mark)

Show Answers Only

a.    `98`

b.    `37/43`

c.    `28`

Show Worked Solution

a.    `A= 200-(74 + 12 + 16)= 98`
 

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

c.    `text(Number 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

a.    `text(See Worked Solutions)`

b.    `170`

c.    `text(85%)`

d.    `2/15`

Show Worked Solution

a.

b.    `text(# Accurate readings) = 40 + 130 = 170`
 

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

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

`= 170/200= 85 text(%)`
 

d.    `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 of females in part-time)`

`text(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

a.    `58`

b.    `6/25`

c.    `text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

a.    `text{Critics liked and}\ >= 40 = 102-65 = 37`

`A = 37+31=68`

 
b.    `text{Critics did not like and < 40} = 175-65-37-31= 42`

`P text{(not like and  < 40)}= 42/175= 6/25`
 

c.    `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`

Probability, STD2 S2 2011 HSC 25c

At another school, students who use mobile phones were surveyed. The set of data is shown in the table.

2UG 2011 25c

  1. How many students were surveyed at this school?   (1 mark)

  2. Of the female students surveyed, one is chosen at random. What is the probability that she uses pre-paid?   (1 mark)

Ten new male students are surveyed and all ten are on a plan. The set of data is updated to include this information.

  1. What percentage of the male students surveyed are now on a plan? Give your answer to the nearest per cent.   (1 mark)

Show Answers Only

a.    `580`

b.    `text(54%)`

c.    `text(42%)`

Show Worked Solution

a.    `text(# Students surveyed)=319+261=580`
 

b.    `Ptext{(Female uses prepaid)}`

`=text(# Females on prepaid)/text(Total females)=172/319=0.53918… ~~ 54\text{% (nearest %)}`
 

c.    `text(% Males on plan)` `=text(# Males on plan + 10)/text(Total males + 10)=(103+10)/(261+10)`
  `=113/271=0.4169…=42\text{%  (nearest %)}`

Probability, STD2 S2 2009 HSC 8 MC

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.
 

2UG-2009-8MC 

What percentage of the women surveyed supported `Team\ B`, correct to the nearest percent?

  1. 23%
  2. 45%
  3. 47%
  4. 55%
Show Answers Only

`D`

Show Worked Solution

`text(% Women for)\ Team\ B= (text(# Women for)\ Team\ B)/text(# Women surveyed)= 90/165= 54.54\ … %`

`=>  D`

Probability, STD2 S2 2010 HSC 12 MC

A group of 347 people was tested for flu and the results were recorded. The flu test  results are not always accurate.

2010 12 MC

A person is selected at random from the tested group.

What is the probability that their test result is accurate, to the nearest per cent?

  1.    21%
  2.    22%
  3.    95%
  4.    96%
Show Answers Only

`C`

Show Worked Solution
`P\ text((Test accurate))` `=text(Accurate readings)/text(Total tested)`
  `=(72+256)/347=94.5244…%`

`=>  C`

Probability, STD2 S2 2013 HSC 10 MC

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

One of these students is selected at random.

 What is the probability that this student is male and living with his parent(s)?

  1. 31%
  2. 40%
  3. 56%
  4. 77%
Show Answers Only

`A`

Show Worked Solution

`text(Number of males living with their parents is = 155)`.

`:.\ P=155/505xx 100=30.693%…approx 31%`

`=>\ A`