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Probability, STD2 2026 HSC 32

At a particular school, the probability that a student walks to school is \(\dfrac{3}{10}\) and the probability that a student wears the correct school uniform is \(\dfrac{9}{10}\).

The probability tree shows this information.
 

  1. One student at the school is selected at random.
  2. What is the probability that this student walks to school and is not wearing the correct school uniform?   (1 mark)
  3. Two students at the school are selected at random.
  4. What is the probability that at least one of these students walks to school and is not wearing the correct school uniform?   (2 marks)
Show Answers Only

a.    \(\dfrac{3}{100}\)

b.    \(\dfrac{591}{10\ 000}\)

Show Worked Solution

a.    \(P(\text{W and NC})=\dfrac{3}{10}\times\dfrac{1}{10}=\dfrac{3}{100}\)
 

b.    \(\text{Let E = student walks and is not in correct uniform}\)

\(P(\text{E})=\dfrac{3}{100}\ \ \Rightarrow\ \ P(\text{not E})=1-\dfrac{3}{100}=\dfrac{97}{100}\)

\(\text{“At least one E” is the complement of “neither student is E”.}\)

\(P(\text{at least one E})\) \(=1-P(\text{not E})\times P(\text{not E})\)
  \(=1-\dfrac{97}{100}\times\dfrac{97}{100}\)
  \(=1-\dfrac{9409}{10\ 000}\)
  \(=\dfrac{591}{10\ 000}\)

Probability, STD2 S2 2022 HSC 17

The numbers 1, 2, 3, 4, 5 and 6 are each written on separate cards.

Amy has cards 1, 3 and 5 and Bob has cards 2, 4 and 6.

They play a game in which each person randomly chooses one of their own cards and compares it with the other person's card. The person with the higher card wins.

  1. A partially completed tree diagram is shown.
     
           
     
    Complete the tree diagram and find the probability that Bob wins.   (2 marks)
     
  2. Suppose Amy and Bob play this game 30 times.
  3. How many times would Bob be expected to win?   (1 mark)

Show Answers Only
  1. `6/9`
  2. `20`
Show Worked Solution

a.   
       

`P(text{Bob wins}) = 6/9=2/3`
  

b.    `text{Expected wins}=2/3 xx 30=20`

Probability, STD2 S2 2021 HSC 11 MC

There are 8 chocolates in a box. Three have peppermint centres (P) and five have caramel centres (C).

Kim randomly chooses a chocolate from the box and eats it. Sam then randomly chooses and eats one of the remaining chocolates.

A partially completed probability tree is shown.
 

What is the probability that Kim and Sam choose chocolates with different centres?

  1. `\frac{15}{64}`
  2. `\frac{15}{56}`
  3. `\frac{15}{32}`
  4. `\frac{15}{28}`
Show Answers Only

`D`

Show Worked Solution

♦♦ Mean mark 35%.

 

`Ptext{(different centres)}` `= P text{(PC)} + P text{(CP)}`
  `=\frac{3}{8}xx \frac{5}{7} + \frac{5}{8}xx \frac{3}{7}`
  `= \frac{15}{56} + \frac{15}{56}`
  `= \frac{15}{28}`

 
`=> D`

Probability, STD2 S2 2019 HSC 25

A bowl of fruit contains 17 apples of which 9 are red and 8 are green.

Dennis takes one apple at random and eats it. Margaret also takes an apple at random and eats it.

By drawing a probability tree diagram, or otherwise, find the probability that Dennis and Margaret eat apples of the same colour.   (3 marks)

Show Answers Only

`8/17`

Show Worked Solution

`P(text(same colour))` `= P(R R) + P(GG)`
  `= 9/17 xx 8/16 + 8/17 xx 7/16`
  `= 72/272 + 56/272= 8/17`
♦♦ Mean mark 35%.

Probability, STD2 S2 2018 HSC 30d

A game consists of two tokens being drawn at random from a barrel containing 20 tokens. There are 17 tokens labelled 10 cents and 3 tokens labelled $2. The player wins the total value of the two tokens drawn.

Complete the probability tree by writing the missing probabilities in the boxes.   (2 marks)

 


Show Answers Only

Show Worked Solution

Probability, STD2 S2 EQ-Bank 31

A game consists of two tokens being drawn at random from a barrel containing 20 tokens. There are 17 red tokens and 3 black tokens. The player keeps the two tokens drawn.

  1.  Complete the probability tree by writing the missing probabilities in the boxes.   (2 marks)
     
     
         

  2.  What is the probability that a player draws at least one red token. Give your answer in exact form.   (2 marks)

Show Answers Only

a.   
           

b.    `187/190`

Show Worked Solution
a.    

 

b.    `P(text(at least one red))`

`= 1-P(BB)`

`= 1-3/20 xx 2/19= 187/190`

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 30b

On a tray there are 12 hard‑centred chocolates `(H)` and 8 soft‑centred chocolates `(S)`. Two chocolates are selected at random. A partially completed probability tree is shown.
 

What is the probability of selecting one of each type of chocolate?   (3 marks)

Show Answers Only

`48/95`

Show Worked Solution

`Ptext{(one of each type)}`

♦ Mean mark 45%.

`= P(HS) + P(SH)`

`= (12/20 xx 8/19) + (8/20 xx 12/19)`

`= 24/95 + 24/95= 48/95`

Probability, STD2 S2 2006 HSC 28a

On a bridge, the toll of $2.50 is paid in coins collected by a machine. The machine only accepts two-dollar coins, one-dollar coins and fifty-cent coins.

  1. List the different combinations of coins that could be used to pay the $2.50 toll.   (1 mark)

  2. Jill has three two-dollar coins, six one-dollar coins and two fifty-cent coins. She selects two coins at random.

     

    What is the probability that she selects exactly $2.50?   (3 marks)

  3. At the end of a day, the machine contains `x` two-dollar coins, `y` one-dollar coins and `w` fifty-cent coins.

     

    Write an expression for the total value of coins in dollars in the machine.   (1 mark)

Show Answers Only

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

b.    `6/55`

c.    `$(2x + y + 0.5w)`

Show Worked Solution

a.    `text(Combinations for $2.50)`

`$2, 50c`

`$1, $1, 50c`

`$1, 50c, 50c, 50c`

`50c, 50c, 50c, 50c, 50c`
  

b.    `3 xx $2, 6 xx $1, 2 xx 50c`

`P($2.50)` `= (3/11 xx 2/10) + (2/11 xx 3/10)`
  `= 6/110 + 6/110= 6/55`

 

c.    `text(Total value) = $ (2x + y + 0.5w)`

Probability, STD2 S2 2005 HSC 23c

Moheb owns five red and seven blue ties. He chooses a tie at random for himself and puts it on. He then chooses another tie at random, from the remaining ties, and gives it to his brother.

  1. What is the probability that Moheb chooses a red tie for himself?   (1 mark)

Copy the tree diagram into your writing booklet.
 

2UG-2005-23c
 

  1. Complete your tree diagram by writing the correct probability on each branch.   (2 marks)
  2. Calculate the probability that both of the ties are the same colour.   (2 marks)

Show Answers Only

a.    `5/12`

b.

c.    `31/66`

Show Worked Solution

a.    `P(R)= (#\ text(red ties))/(#\ text(total ties))=5/12`

 

b.    

 

c.    `Ptext((same colour))`

`= P(text(RR)) + P(text(BB))`

`= 5/12 × 4/11\ \ +\ \ 7/12 × 6/11= 20/132 + 42/132= 31/66`

Probability, STD2 S2 2007 HSC 25c

In a stack of 10 DVDs, there are 5 rated PG, 3 rated G and 2 rated M.

  1. A DVD is selected at random. What is the probability that it is rated M?   (1 mark)

Grant chooses two DVDs at random from the stack. Copy or trace the tree diagram into your writing booklet.
 

  1. Complete the tree diagram by writing the correct probability on each branch.   (2 marks)

  2. Calculate the probability that Grant chooses two DVDs with the same rating.   (2 marks)

Show Answers Only

a.    `1/5`

b.  

c.    `14/45`

Show Worked Solution

a.    `text(5 PG, 3 G, 2 M)`

`P text{(M)} = 2/10 = 1/5` 

 

b.     

 

c.    `P text{(same rating)}`

`= P text{(PG, PG)} + P text{(G, G)} + P text{(M, M)}`

`= (1/2 xx 4/9) + (3/10 xx 2/9) + (1/5 xx 1/9)`

`= 2/9 + 1/15 + 1/45= 14/45`

Probability, STD2 S2 2008 HSC 25b

In a drawer there are 30 ribbons. Twelve are blue and eighteen are red.

Two ribbons are selected at random.

  1. Copy and complete the probability tree diagram.   (1 mark)
     

     
  2. What is the probability of selecting a pair of ribbons which are the same colour?   (2 marks)

Show Answers Only

a.

b.    `73/145`

Show Worked Solution

a. 

b.    `Ptext{(same colour)}`

`=\ text{P(BB) + P(RR)}`

`= 12/30 xx 11/29 + 18/30 xx 17/29`

`= 132/870 + 306/870= 73/145` 

Probability, STD2 S2 2014 HSC 28c

A fair coin is tossed three times. Using a tree diagram, or otherwise, calculate the probability of obtaining two heads and a tail in any order.   (2 marks)

Show Answers Only

`3/8`

Show Worked Solution

 
`P text{(2 heads, 1 tail)}`

`= P(HHT) + P(HTH) + P(THH)`

`= (1/2 xx 1/2 xx 1/2) + (1/2 xx 1/2 xx 1/2) + (1/2 xx 1/2 xx 1/2)`

`= 1/8 + 1/8 + 1/8= 3/8`

Probability, STD2 S2 2014 HSC 19 MC

Jaz has 2 bags of apples.
  

Bag A contains 4 red apples and 3 green apples. 

Bag B contains 3 red apples and 1 green apple.
  

Jaz chooses an apple from one of the bags. 

Which tree diagram could be used to determine the probability that Jaz chooses a red apple?
 

2014 19 mc1

2014 19 mc2

Show Answers Only

`A`

Show Worked Solution

`text(The tree diagram needs to identify 2 separate events.)`

`text(1st event – which bag is chosen)`

`text(2nd event – choosing a red apple from a particular bag)`

`=>  A`

Probability, STD2 S2 2013 HSC 30b

In a class there are 15 girls (G) and 7 boys (B). Two students are chosen at random to be class representatives.

  1. Complete the tree diagram below.   (2 marks)
     
     

  2. What is the probability that the two students chosen are of the same gender?   (2 marks)

Show Answers Only

 a.   

b.    `6/11`

Show Worked Solution

a.  

b.      `Ptext{(same gender)}` `=P(G,G) + P(B,B)`
    `=(15/22 xx 14/21) + (7/22 xx 6/21)`
    `=210/462 + 42/462`
    `=252/462=6/11`
♦ Mean mark (b) 40%.

Probability, STD2 S2 2010 HSC 20 MC

Lou and Ali are on a fitness program for one month. The probability that Lou will finish the program successfully is 0.7 while the probability that Ali will finish successfully is 0.6. The probability tree shows this information

 

What is the probability that only one of them will be successful ?

  1. `0.18`
  2. `0.28`
  3. `0.42`
  4. `0.46`
Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ Ptext{(Lou successful)}=P(L) = 0.7, \ P(\text{not}\ L) = 0.3`

`text(Let)\ \ Ptext{(Ali successful)}=P(A) = 0.6, \ P(\text{not}\ A) = 0.4`

`P text{(only 1 successful)}` `=P(L)xxP(text(not)\ A)+P(text(not)\ L)xxP(A)`
  `=(0.7xx0.4)+(0.3xx0.6)`
  `=0.28+0.18=0.46`

 
`=>  D`

♦ Mean mark 48%.

Probability, STD2 S2 2012 HSC 27e

A box contains 33 scarves made from two different fabrics. There are 14 scarves made from silk (S) and 19 made from wool (W).
Two girls each select, at random, a scarf to wear from the box.

  1. Complete the probability tree diagram below.   (2 marks) 
      
       

  2. Calculate the probability that the two scarves selected are made from silk.   (1 mark)

  3. Calculate the probability that the two scarves selected are made from different fabrics.   (2 marks)

Show Answers Only

a. 

   
 

b.    `P\ text{(2 silk)}= 91/528`

c.    `P\ text{(different fabrics)}= 133/264`

Show Worked Solution

a. 

♦ Mean mark (a) 43%.
b.    `P\ text{(2 silk)}` `= P(S_1) xx P(S_2)`
  `= 14/33 xx 13/32= 91/528`

 

c.    `P\ text{(different)}` `= P (S_1,W_2) + P(W_1,S_2)`
  `= (14/33 xx 19/32) + (19/33 xx 14/32)`
  `= 532/1056= 133/264`
♦ Mean mark (c) 41%.
MARKER’S COMMENT: In better responses, students multiplied along the branches and then added these two results together