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Probability, STD1 S2 2025 HSC 8 MC

A spinner made up of 4 colours is spun 100 times. The frequency histogram shows the results.
 

Which of these spinners is most likely to give the results shown?
 

Show Answers Only

\(A\)

Show Worked Solution
\(P(\text{White})\) \(=\dfrac{50}{100}=\dfrac{1}{2}\)
\(P(\text{Red})\) \(=\dfrac{25}{100}=\dfrac{1}{4}\)  
\(P(\text{Yellow})\) \(=\dfrac{15}{100}=\dfrac{3}{20}\)
\(P(\text{Green})\) \(=\dfrac{10}{100}=\dfrac{2}{20}=\dfrac{1}{10}\)

 
\(\text{Eliminate Options B and D as white}\ \neq \dfrac{1}{2}\ \text{of spinner.}\)

\(\text{Eliminate Option C as red}\ \neq \dfrac{1}{4}\ \text{of spinner.}\)

\(\Rightarrow A\)

Probability, STD1 S2 2024 HSC 17

A wheel is shown with the numbers 0 to 19 marked.

A game is played where the wheel is spun until it stops.

When the wheel stops, a pointer points to the winning number. Each number is equally likely to win.
 

  1. List all the even numbers on the wheel that are greater than 7.   (1 mark)

  2. What is the probability that the winning number is NOT an even number greater than 7?   (2 marks)

Show Answers Only

a.    \(8\ ,\ 10\ ,\ 12\ ,\ 14\ ,\ 16\ ,\ 18\)

b.    \(0.7\)

Show Worked Solution

a.    \(8\ ,\ 10\ ,\ 12\ ,\ 14\ ,\ 16\ ,\ 18\ \text{(6 numbers)}\)
 

b.   \(\text{Total numbers = 20}\)

\(\text{Numbers not even and > 7}\ = 20-6=14\ \text{numbers}\)

\(P\text{(not even and > 7)}\ =\dfrac{14}{20}=0.7\)

♦♦ Mean mark (b) 32%.

Probability, STD1 S2 2023 HSC 8 MC

Four cards marked with the numbers 1, 2, 3 and 4 are placed face down on a table.

One card is turned over as shown.

What is the probability that the next card turned over is marked with an odd number?

  1. \(\dfrac{1}{4}\)
  2. \(\dfrac{1}{3}\)
  3. \(\dfrac{2}{4}\)
  4. \(\dfrac{2}{3}\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Sample space} =1,3,4\)

\(P(\text{odd})=\dfrac{2}{3}\)

  
\(\Rightarrow D\)

Probability, STD2 S2 2023 HSC 8 MC

A game involves throwing a die and spinning a spinner.

The sum of the two numbers obtained is the score.

The table of scores below is partially completed.
 

What is the probability of getting a score of 7 or more?

  1. `1/6`
  2. `1/4`
  3. `5/18`
  4. `5/12`

Show Answers Only

`D`

Show Worked Solution

`Ptext{(score 7+)} = 10/24=5/12`

`=>D`

Probability, STD1 S2 2022 HSC 17

Each number from 1 to 30 is written on a separate card. The 30 cards are shuffled. A game is played where one of these cards is selected at random. Each card is equally likely to be selected.

Ezra is playing the game, and wins if the card selected shows an odd number between 20 and 30.

  1. List the numbers which would result in Ezra winning the game.   (1 mark)

  2. What is the probability that Ezra does NOT win the game?   (2 marks)

Show Answers Only

a.    `21, 23, 25, 27, 29`

b.    `Ptext{(not win)} = 5/6`

Show Worked Solution

a.    `21, 23, 25, 27, 29`
 

b.     `Ptext{(not win)}` `=1-Ptext{(win)}`
    `=1-5/30=25/30=5/6`


♦ Mean mark 51%.

Probability, STD1 S2 2022 HSC 3 MC

A jar contains 12 red, 10 black and 13 white lollies.

Alex picks out a red lolly and eats it. He then randomly picks a second lolly.

What is the probability that the second lolly is also red?

  1. `(11)/(34)`
  2. `(11)/(35)`
  3. `(12)/(34)`
  4. `(12)/(35)`
Show Answers Only

`A`

Show Worked Solution
`P(E)` `=text{favourable outcomes}/text{total outcomes}`  
  `=(12-1)/((12-1)+10+13)=11/34`  

 
`=>A`

Probability, STD1 S2 2021 HSC 20

In a bag, there are six playing cards, 2, 4, 6, 8, Queen and King. The Queen and King are known as picture cards.

Two of these cards are chosen randomly. All the possible outcomes are shown.
 

   
 

  1. What is the probability that the two cards chosen include one or both picture cards?   (1 mark)

  2. What is the probability that the two cards chosen do NOT include any picture cards?   (1 mark)

Show Answers Only
  1. `9/15=3/5`
  2. `6/15=2/5`
Show Worked Solution

a.   `P text{(at least 1 picture card)} = 9/15=3/5`

 

b.    `P text{(no picture card)}` `= 1-9/15`
    `= 6/15=2/5`

Probability, STD1 S2 2020 HSC 26

Barbara plays a game of chance, in which two unbiased six-sided dice are rolled. The score for the game is obtained by finding the difference between the two numbers rolled. For example, if Barbara rolls a 2 and a 5, the score is 3.

The table shows some of the scores.
 


 

  1. Complete the six missing values in the table to show all possible scores for the game.   (1 mark)
  2. What is the probability that the score for a game is NOT 0?   (2 marks)

Show Answers Only

 a.

b.    `frac{5}{6}`

Show Worked Solution

a.     

♦ Mean mark part (b) 47%.
b.      `Ptext{(not zero)}` `= frac{text(numbers) ≠ 0}{text(total numbers)}`
    `= frac{30}{36}= frac{5}{6}`

 
\(\text{Alternate solution (b)}\)

b.      `Ptext{(not zero)}` `= 1-Ptext{(zero)}`
    `= 1-frac{6}{36}`
    `= frac{5}{6}`

Probability, STD2 S2 2011 HSC 26a

The two spinners shown are used in a game.

2UG 2011 26a1

Each arrow is spun once. The score is the total of the two numbers shown by the arrows.
A table is drawn up to show all scores that can be obtained in this game.

2UG 2011 26a2

  1. What is the value of `X` in the table?   (1 mark)

  2. What is the probability of obtaining a score less than 4?   (1 mark)

  3. On Spinner `B`, a 2 is obtained. What is the probability of obtaining a score of 3?   (1 mark)

Show Answers Only

a.    `5`

b.    `1/2`

c.    `2/3`

Show Worked Solution

a.    `X=3+2=5`

b.    `P(text{score}<4)=6/12=1/2`

c.    `P(3)=2/3`

Probability, STD2 S2 2018 HSC 9 MC

An experiment has three distinct outcomes, A, B and C.

Outcome A occurs 50% of the time. Outcome B occurs 23% of the time.

What is the expected number of times outcome C would occur if the experiment is conducted 500 times?

  1. 115
  2. 135
  3. 250
  4. 365
Show Answers Only

`B`

Show Worked Solution

`text(Expectation of outcome)\ C`

`= 1-0.5-0.23= 0.27`
 

`:.\ text(Expected times)\ C\ text(occurs)`

`= 0.27 xx 500= 135`

`=> B`

Probability, STD2 S2 2017 HSC 15 MC

The faces on a twenty-sided die are labelled  $0.05, $0.10, $0.15, … , $1.00.

The die is rolled once.

What is the probability that the amount showing on the upper face is more than 50 cents but less than 80 cents?

  1. `1/4`
  2. `3/10`
  3. `7/20`
  4. `1/2`
Show Answers Only

`A`

Show Worked Solution

`text(Possible faces that satisfy are:)`

♦ Mean mark 50%.

`55text(c),60text(c),65text(c),70text(c),75text(c)`

`:.\ text(Probability)= 5/20= 1/4`
  

`=>A`

Probability, STD2 S2 2016 HSC 28c

A cricket team is about to play two matches. The probability of the team having a win, a loss or a draw is 0.7, 0.1 and 0.2 respectively in each match. The possible results in the two matches are displayed in the probability tree diagram.
  

2ug-2016-hsc-q28_2

  1. What is the probability of the team having a win and a draw, in any order?   (2 marks)

  2. Paul claims that 1.4 is the probability of the team winning both matches.

     

    Give one reason why this is NOT correct.   (1 mark)

Show Answers Only

i.    `0.28`

ii.   `text(Probabilities cannot exceed 1.)`

Show Worked Solution

i.    `P(W\ text(and)\ D)`

`= P(W,D) + P(D,W)`

`= 0.7 xx 0.2 + 0.2 xx 0.7= 0.28`

♦ Mean mark (i) 45%.
♦ Mean mark (ii) 49%.

 
ii.   `text(Probabilities cannot exceed 1.)`

Probability, STD2 S2 2015 HSC 26e

The table shows the relative frequency of selecting each of the different coloured jelly beans from packets containing green, yellow, black, red and white jelly beans.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Colour} \rule[-1ex]{0pt}{0pt} & \textit{Relative frequency} \\
\hline
\rule{0pt}{2.5ex} \text{Green} \rule[-1ex]{0pt}{0pt} & 0.32 \\
\hline
\rule{0pt}{2.5ex} \text{Yellow} \rule[-1ex]{0pt}{0pt} & 0.13 \\
\hline
\rule{0pt}{2.5ex} \text{Black} \rule[-1ex]{0pt}{0pt} & 0.14 \\
\hline
\rule{0pt}{2.5ex} \text{Red} \rule[-1ex]{0pt}{0pt} &  \\
\hline
\rule{0pt}{2.5ex} \text{White} \rule[-1ex]{0pt}{0pt} & 0.24 \\
\hline
\end{array}

  1. What is the relative frequency of selecting a red jelly bean?   (1 mark)

  2. Based on this table of relative frequencies, what is the probability of NOT selecting a black jelly bean?   (1 mark)

Show Answers Only

a.    \(0.17\)

b.    \(0.86\)

Show Worked Solution

a.    \(\text{Relative frequency of red}\)

\(= 1-(0.32 + 0.13 + 0.14 + 0.24)= 1-0.83= 0.17\)

 

b.    \(P\text{(not selecting black)}\)

\(= 1-P\text{(selecting black)}= 1-0.14= 0.86\)

Probability, STD2 S2 2005 HSC 11 MC

The diagram shows a spinner.
 


 

The arrow is spun and will stop in one of the six sections.

What is the probability that the arrow will stop in a section containing a number greater
than 4?

  1. `2/5`
  2. `2/3`
  3. `1/3`
  4. `1/2`
Show Answers Only

`D`

Show Worked Solution

`P\ text((number greater than 4))`

`= P(7) + P (9)`

`= 2/6 + 1/6= 1/2`

`=>  D`

Probability, STD2 S2 2006 HSC 10 MC

Kay randomly selected a marble from a bag of marbles, recorded its colour and returned it to the bag. She repeated this process a number of times.
  


  

Based on these results, what is the best estimate of the probability that Kay will choose a green marble on her next selection?

  1. `5/24`
  2. `1/24`
  3. `1/6`
  4. `1/5`
Show Answers Only

`C`

Show Worked Solution
`text{P(Green)}` `= text(# Green chosen) / text(Total Selections)`
  `= 4/24= 1/6`

`=>  C`

Probability, STD2 S2 2006 HSC 1 MC

The probability of an event occurring is `9/10.`

Which statement best describes the probability of this event occurring?

  1. The event is likely to occur.
  2. The event is certain to occur.
  3. The event is unlikely to occur.
  4. The event has an even chance of occurring.
Show Answers Only

`A`

Show Worked Solution

`text(The event is highly likely to occur but not certain.)`

`=>  A`

Probability, STD2 S2 2005 HSC 23a

There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
 

  1. Justine claims that each of her friends is equally likely to win first prize.

     

    Give a reason why Justine’s statement is NOT correct.   (1 mark)

  2. What is the probability that first prize is NOT won by Khalid or Herman?   (2 marks)

Show Answers Only

a.    `text(The claim is incorrect because each of her friends)`

`text(bought a different number of tickets and therefore)`

`text(their chances of winning are different.)`

b.    `69/100`

Show Worked Solution

a.    `text(The claim is incorrect because each of her friends bought)`

`text(a different number of tickets and therefore their chances of)`

`text(winning are different.)`

 

b.    `text(Number of tickets not sold to K or H)= 45 + 10 + 14= 69` 

`:.\ text(Probability 1st prize NOT won by K or H)= 69/100`

Probability, STD2 S2 2004 HSC 18 MC

Two dice are rolled. What is the probability that only one of the dice shows a six?

  1. `5/36`
  2. `1/6`
  3. `5/18`
  4. `11/36`
Show Answers Only

`C`

Show Worked Solution

`text(Method 1: Using an array)`

`P text{(only 1 six)}=10/36=5/18`

\begin{align}
\textbf{Die B }
\begin{array}{c}
\textbf{Die A}  \\
\begin{array}{c|c|c|c|c|c|c}
\ & 1 & 2 & 3 & 4 & 5 & 6  \\
\hline
\ 1 & 1,1  & 1,2 & 1,3 & 1,4 & 1,5 & \fcolorbox{red}{white}{1,6} \\
\hline
\ 2 & 2,1 & 2,2 & 2,3 & 2,4 & 2,5 & \fcolorbox{red}{white}{2,6}  \\
\hline
\ 3 & 3,1 & 3,2 & 3,3 & 3,4 & 3,5 & \fcolorbox{red}{white}{3,6}  \\
\hline
\ 4 & 4,1 & 4,2 & 4,3 & 4,4 & 4,5 & \fcolorbox{red}{white}{4,6}  \\
\hline
\ 5 & 5,1 & 5,2 & 5,3 & 5,4 & 5,5 & \fcolorbox{red}{white}{5,6}  \\
\hline
\ 6 & \fcolorbox{red}{white}{6,1} & \fcolorbox{red}{white}{6,2} & \fcolorbox{red}{white}{6,3} & \fcolorbox{red}{white}{6,4} & \fcolorbox{red}{white}{6,5} & 6,6  \\
\end{array}
\end{array}
\end{align}

 

`text(Method 2:)`

`text{P (Only 1 six)}`

`= P text{(6, not 6)} + P text{(not 6, 6)}`

`= 1/6 xx 5/6 + 5/6 xx 1/6`

`= 10/36= 5/18`

`=>  C`

Probability, STD2 S2 2005 HSC 3 MC

Four radio stations reported the probability of rain as shown in the table.
 

Which radio station reported the highest probability of rain?

  1. `text(2AT)`
  2. `text(2BW)`
  3. `text(2CZ)`
  4. `text(2DL)`
Show Answers Only

`D`

Show Worked Solution

`text(Converting all probabilities to decimals)`

`2AT` `= 0.53`
`2BW` `= 0.17`
`2CZ` `= 0.52`
`2DL` `= 0.60`

  
`=> D`

Probability, STD2 S2 2004 HSC 1 MC

Which fraction is equal to a probability of `text(25%)`?

  1. `1/25`
  2. `1/4`
  3. `1/3`
  4. `1/2`
Show Answers Only

`B`

Show Worked Solution

`P=25/100=1/4`

`=> B`

Probability, STD2 S2 2007 HSC 25a

Give an example of an event that has a probability of exactly  `3/4`.   (1 mark)

Show Answers Only

`text(Choosing a red ball out of a bag that)`

`text(contains 3 red balls and 1 green ball.)`

`text{(An infinite amount of examples are}`

`text{possible)}`

Show Worked Solution

`text(Choosing a red ball out of a bag that contains)`

`text(3 red balls and 1 green ball.)`

`text{(An infinite amount of examples are}`

`text{possible)}`

Probability, STD2 S2 2008 HSC 16 MC

A bag contains some marbles. The probability of selecting a blue marble at random from this bag is  `3/8`.

Which of the following could describe the marbles that are in the bag?

  1. `3`  blue,  `8`  red
  2. `6`  blue,  `11`  red
  3. `3`  blue,  `4`  red,  `4`  green
  4. `6`  blue,  `5`  red,  `5`  green 
Show Answers Only

`D`

Show Worked Solution

`P(B) = 3/8`

`text(In)\ A,\ \ ` `P(B) = 3/11`
`text(In)\ B,\ \ ` `P(B) = 6/17 `
`text(In)\ C,\ \ ` `P(B) = 3/11`
`text(In)\ D,\ \ ` `P(B) = 6/16 = 3/8`

`=>  D`

Probability, STD2 S2 2011 HSC 15 MC

An unbiased coin is tossed 10 times.

A tail is obtained on each of the first 9 tosses.

What is the probability that a tail is obtained on the 10th toss?

  1. `1/2^10`
  2. `1/2`
  3. `1/10`
  4. `9/10`
Show Answers Only

`B`

Show Worked Solution

`text(Each toss is an independent event and has an even chance)`

`text(of being a head or tail.)`

`=> B`

Probability, STD2 S2 2010 HSC 23c

On Saturday, Jonty recorded the colour of T-shirts worn by the people at his gym. The results are shown in the graph.

 

  1. How many people were at the gym on Saturday? (Assume everyone was wearing a T-shirt).   (1 mark)

  2. What is the probability that a person selected at random at the gym on Saturday, would be wearing either a blue or green T-shirt?   (1 mark)

Show Answers Only

a.    `34`

b.    `15/34`

Show Worked Solution

a.    `text(# People)=5+15+10+3+1=34`
  

b.    `P (B\ text{or}\ G)=P(B)+P(G)=5/34+10/34=15/34`

Probability, STD2 S2 2009 HSC 1 MC

A newspaper states: ‘It will most probably rain tomorrow.’

Which of the following best represents the probability of an event that will most probably occur?   

  1. `33 1/3 text(%)` 
  2.  `text(50%)` 
  3. `text(80%)` 
  4. `text(100%)` 
Show Answers Only

`C`

Show Worked Solution

`text(Probably) =>\ text(likelihood > 50%)`

`text(However 100% = certainty)`

`:.\ text(80% is the answer)`

`=> C`

Probability, STD2 S2 2011 HSC 2 MC

Which of the following could be the probability of an event occurring?

  1. `1` 
  2. `6/5` 
  3. `1.27` 
  4. `text(145%)` 
Show Answers Only

`A`

Show Worked Solution

`text(Probabilities must lie between 0 and 1 inclusive.)`

`=>A`

Probability, STD2 S2 2010 HSC 8 MC

A bag contains red, green, yellow and blue balls.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Colour} \rule[-1ex]{0pt}{0pt} & \textit{Probability} \\
\hline
\rule{0pt}{2.5ex} \text{Red} & \dfrac{1}{3} \\
\hline
\rule{0pt}{2.5ex} \text{Green}  & \dfrac{1}{4} \\
\hline
\rule{0pt}{2.5ex} \text{Yellow}  & \text{?} \\
\hline
\rule{0pt}{2.5ex} \text{Blue}  & \dfrac{1}{6} \\
\hline
\end{array}

The table shows the probability of choosing a red, green, or blue ball from the bag.

If there are 12 yellow balls in the bag, how many balls are in the bag altogether

  1. 16
  2. 36
  3. 48
  4. 60
Show Answers Only

\(C\)

Show Worked Solution
\(P(R)+P(G)+P(Y)+P(B)\) \(=1\)
\(\dfrac{1}{3}+\dfrac{1}{4}+P(Y)+\dfrac{1}{6}\) \(=1\)

 

\(P(Y)\) \(= 1-(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{6})\)
  \(=1-\dfrac{9}{12}\)
  \(=\dfrac{1}{4}\)

 

\(P(Y)\) \(=\dfrac{\text{Yellow balls}}{\text{Total balls}}\)
\(\dfrac{1}{4}\) \(=\dfrac{12}{\text{Total balls}}\)

  
\(\therefore\ \text{ Total balls}=48\)

\(\Rightarrow C\)

Probability, STD2 S2 2012 HSC 12 MC

Two unbiased dice, each with faces numbered 1, 2, 3, 4, 5, 6, are rolled. 

What is the probability of a 6 appearing on at least one of the dice? 

  1. `1/6`  
  2. `11/36` 
  3. `25/36`  
  4. `5/6`  
Show Answers Only

`B`

Show Worked Solution

`text(Method 1: Using an array`

`P text{(at least 1 six)}=11/36`

\begin{align}
\textbf{Die B }
\begin{array}{c}
\textbf{Die A}  \\
\begin{array}{c|c|c|c|c|c|c}
\ & 1 & 2 & 3 & 4 & 5 & 6  \\
\hline
\ 1 & 1,1  & 1,2 & 1,3 & 1,4 & 1,5 & \fcolorbox{red}{white}{1,6} \\
\hline
\ 2 & 2,1 & 2,2 & 2,3 & 2,4 & 2,5 & \fcolorbox{red}{white}{2,6}  \\
\hline
\ 3 & 3,1 & 3,2 & 3,3 & 3,4 & 3,5 & \fcolorbox{red}{white}{3,6}  \\
\hline
\ 4 & 4,1 & 4,2 & 4,3 & 4,4 & 4,5 & \fcolorbox{red}{white}{4,6}  \\
\hline
\ 5 & 5,1 & 5,2 & 5,3 & 5,4 & 5,5 & \fcolorbox{red}{white}{5,6}  \\
\hline
\ 6 & \fcolorbox{red}{white}{6,1} & \fcolorbox{red}{white}{6,2} & \fcolorbox{red}{white}{6,3} & \fcolorbox{red}{white}{6,4} & \fcolorbox{red}{white}{6,5} & \fcolorbox{red}{white}{6,6}  \\
\end{array}
\end{array}
\end{align}

 

`text(Method 2: Using )P text{(E)} = 1-P\text{(not E)}`
  

`P text{(at least 1 six)}`

`=1-P text{(no six)} xx P text{(no six)} `

`=1-5/6 xx 5/6=11/36`

`=>  B`

♦♦♦ Mean mark 25%
COMMENT: The term “at least” should flag that calculating the probability of `1-P text{(event not happening)}` is likely to be the most efficient way to solve.

Probability, STD2 S2 2013 HSC 18 MC

Two unbiased dice, each with faces numbered  1, 2, 3, 4, 5, 6,  are rolled.

What is the probability of obtaining a sum of 6?

  1. `1/6`
  2. `1/12`
  3. `5/12`
  4. `5/36`
Show Answers Only

`D`

Show Worked Solution

`text(Total outcomes)=6xx6=36`

`text{Outcomes that sum to 6}=text{(1,5) (5,1) (2,4) (4,2) (3,3)} =5`

`:.\ P\text{(sum of 6)} =5/36`

`=> D`

♦♦ Mean mark 35%.

Probability, STD2 S2 2013 HSC 1 MC

Which of the following events would be LEAST likely to occur?

  1. Tossing a fair coin and obtaining a head
  2. Rolling a standard six-sided die and obtaining a 3
  3. Randomly selecting the letter 'G' from the 26 letters of the alphabet
  4. Winning first prize in a raffle of 100 tickets in which you have 4 tickets
Show Answers Only

`C`

Show Worked Solution

`P(A)=1/2,\ \ P(B)=1/6`

`P(C)=1/26,\ \ P(D)=4/100=1/25`
  

`=> C`