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Probability, SMB-015

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 at least one soft-centred chocolate?  (3 marks)

Show Answers Only

`62/95`

Show Worked Solution

`P(text{at least one}\ S)`

`= 1-P(HH)`

`= 1-(12/20 xx 11/19)`

`= 1-33/95`

`= 62/95`

♦ Mean mark 45%.

Probability, SMB-014

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
  1.  
  2. `187/190`
Show Worked Solution
i.   

 

ii.   `P(text(at least one red))`

`= 1-P(BB)`

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

`= 187/190`

Probability, SMB-013

Beib owns three white and five blue T-shirts. He chooses a T-shirt at random for himself and puts it on. He then chooses another T-shirt at random, from the remaining T-shirts, and gives it to his brother.

  1. What is the probability that Beib chooses a blue T-shirt for himself?  (1 mark)

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

  3. Calculate the probability that both of the T-shirts are the same colour.  (2 marks)

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  1. `5/8`
  2.  
  3. `13/28`
Show Worked Solution

i.    `P(B)= (text(number of blue T-shirts))/(text(total number of T-shirts)) = 5/8`

  
ii.    

       

 
iii. `Ptext((same colour))`

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

`= 3/8 xx 2/7 + 5/8 xx 4/7`

`= 6/56 + 20/56`

`= 13/28`

Probability, SMB-011

Tay-Tay 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.

Tay-Tay chooses one of the bags randomly and with her eyes closed, takes one of the apples. 

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

  2. Determine the probability that Tay-Tay chooses a red apple?   (2 marks) 

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i.    
       

ii.    `37/56`

Show Worked Solution

i.    
       
 

ii.     `Ptext{(red)}` `=1/2 xx 4/7 + 1/2 xx 3/4`
    `=37/56`

Probability, SMB-004 MC

Peter has a bag of marbles. 75% of his marbles are blue.

Peter takes a green marble from his bag and loses it in a game.

If he takes another marble from the bag without looking, what are the chances it is blue?

  1. less than 75%
  2. equal to 75%
  3. greater than 75%
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`C`

Show Worked Solution

`text{Taking marbles two times without replacement → dependent events.}`

`text(When taking the second marble, there will be greater than 75% chance)`

`text(of choosing blue because there are the same amount of  blue marbles)`

`text(to be chosen but 1 less marble of another colour.)`

`=>C`

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)

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`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 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)

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  1. `0.28`
  2. `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 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)

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  1. `1/5`
  2.  
  3. `14/45`
Show Worked Solution

i.    `text(5 PG, 3 G, 2 M)`

`P text{(M)} = 2/10 = 1/5` 

 

ii.   

 

iii.  `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 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)

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  1.  
  2. `6/11`
Show Worked Solution

i.  

ii.    `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 (ii) 40%.

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)

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  1.  
       
     
  2. `P\ text{(2 silk)}= 91/528`
  3. `P\ text{(different fabrics)}= 133/264`
Show Worked Solution

i. 

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

 

iii.  `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 (iii) 41%.
MARKER’S COMMENT: In better responses, students multiplied along the branches and then added these two results together