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Probability, STD2 S2 2006 HSC 25a

Three cards labelled `C`, `J` and `M` can be arranged in any order.

  1. In how many different ways can the cards be arranged?  (1 mark)

  2. What is the probability that the second card in an arrangement is a `J`?  (1 mark)

  3. What is the probability that the last card in an arrangement is not a `C`?  (1 mark)

Show Answers Only
  1. `6`
  2. `1/3`
  3. `2/3`
Show Worked Solution
i.  `text(# Arrangements)` `= 3 xx 2 xx 1`
  `= 6`

 

ii.  `P\ text{(second card is J)}`

`= 1/3`

 

iii.  `P\ text{(last card is not a C)}`

`= 1 – P\ text{(last card is a C)}`

`= 1 – 1/3`

`= 2/3`

Probability, STD2 S2 2008 HSC 18 MC

New car registration plates contain two letters followed by two numerals followed by two more letters eg  AC 12 DC. Letters and numerals may be repeated.

Which of the following expressions gives the correct number of car registration plates that begin with the letter M?

  1.    `26^3 xx 10^2` 
  2.    `25^3 xx 10^2`
  3.    `26^4 xx 10^2`
  4.    `25^4 xx 10^2 `
Show Answers Only

`A`

Show Worked Solution

`text(# Plates beginning with M)`

`= 1 xx 26 xx 10 xx 10 xx 26 xx 26`

`= 26^3 xx 10^2`

`=>  A`

Probability, STD2 S2 2010 HSC 26a

A design of numberplates has a two-digit number, two letters and then another two-digit number, for example

2010 26a1

  1. How many different numberplates are possible using this design?   (1 mark)

Jo can order a numberplate with ‘JO’ in the middle but will have to have randomly selected numbers on either side.

Jo’s birthday is 30 December 1992, so she would like a numberplate with either 

2010 26a2

  1. What is the probability that Jo is issued with one of the numberplates she would like?   (2 marks)

Show Answers Only
  1. `6\ 760\ 000`
  2. `P = 1/5000`
Show Worked Solution
♦ Mean mark 41%
i.    `text(# Combinations)` `=10 xx 10 xx 26 xx 26 xx 10 xx 10`
    `=6\ 760\ 000`

 

♦♦ Mean mark 30%
IMPORTANT: Since the middle letters of “JO” can be guaranteed, the focus becomes purely on the 4 surrounding digits.

ii.   `text(# Possible numberplates)`

`=10 xx 10 xx 10 xx 10`

`=10\ 000`
 

`:.P (30\ text(JO)\ 12) + P (19\ text(JO)\ 92)`
`= 1/(10\ 000) + 1/(10\ 000)`
`= 1/5000`

Probability, STD2 S2 2013 HSC 29c

Mary is designing a website that requires unique logins to be generated.

She plans to generate the logins using two capital letters from the alphabet followed by a series of numerals from 0 to 9 inclusive. All logins will have the same number of numerals. Repetition of letters and numerals is allowed.

What is the minimum number of numerals required for each login so that Mary can generate at least 3 million logins?

Justify your answer with suitable calculations.   (2 marks)

Show Answers Only

 `4`

Show Worked Solution
♦ Mean mark 34%
COMMENT: Students can use their rough working to find an appropriate “number of numerals” where their answer should start.

`text(# Combinations must be > 3 million:)`

`text(If we have 3 numerals,)`

`text(# Combinations)` `=26 xx 26 xx 10 xx 10 xx 10`
  `=676\ 000 < 3\ 000\ 000`

`=> text(need more numeral(s) )`

 

`text(If we have 4 numerals,)`

`text(# Combinations)` `=26 xx 26 xx 10 xx 10 xx 10 xx 10`
  `=6760\ 000 > 3\ 000\ 000`

 

`:.\ text(Minimum number of numerals) = 4`