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Probability, STD2 S2 2025 HSC 29

A bag contains 7 blue lollies and 9 yellow lollies. One lolly is selected at random and eaten. A second lolly is then selected from the remaining lollies in the bag.

Find the probability that the two lollies selected are the same colour.   (2 marks)

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\(P\text{(same colour)}=\dfrac{19}{40}\)

Show Worked Solution

\(P\text{(same colour)}\) \(=P(BB)+P(YY)\)
  \(=\dfrac{7}{16} \times \dfrac{6}{15}+\dfrac{9}{16} \times \dfrac{8}{15}\)
  \(=\dfrac{19}{40}\)

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)

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  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 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 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)

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  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `6/55`
  3. `$(2x + y + 0.5w)`
Show Worked Solution

i.  `text(Combinations for $2.50)`

`$2, 50c`

`$1, $1, 50c`

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

`50c, 50c, 50c, 50c, 50c`

 

ii.  `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`

 

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

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)

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`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`