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Probability, MET2 2023 VCAA 8 MC

A box contains \(n\) green balls and \(m\) red balls. A ball is selected at random, and its colour is noted. The ball is then replaced in the box.

In 8 such selections, where \(n\neq m\), what is the probability that a green ball is selected at least once?

  1. \(8\Bigg(\dfrac{n}{n+m}\Bigg)\Bigg(\dfrac{m}{n+m}\Bigg)^7\)
  2. \(1-\Bigg(\dfrac{n}{n+m}\Bigg)^8\)
  3. \(1-\Bigg(\dfrac{m}{n+m}\Bigg)^8\)
  4. \(1-\Bigg(\dfrac{n}{n+m}\Bigg)\Bigg(\dfrac{m}{n+m}\Bigg)^7\)
  5. \(1-8\Bigg(\dfrac{n}{n+m}\Bigg)\Bigg(\dfrac{m}{n+m}\Bigg)^7\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Let}\ \ X=\ \text{choosing a green ball}\)

\(\text{Pr}(X\geq 1)\) \(=1-\text{Pr}(X=0)\)
  \(=1-\Bigg(\dfrac{m}{n+m}\Bigg)^8\)

 
\(\Rightarrow C\)


♦ Mean mark 49%.

Probability, MET1 2017 VCAA 5

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is `2/5`. Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

  1. What is the probability that Jac does not log on to the computer successfully?   (1 mark)

  2. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form `a/b`, where `a` and `b` are positive integers.   (1 mark)

  3. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form `c/d`, where `c` and `d` are positive integers.   (2 marks)

Show Answers Only

  1. `27/125`
  2. `98/125`
  3. `48/125`

Show Worked Solution

a.  `text(Pr)\ (S^{′} S^{′} S^{′})`

`= (3/5)^3`

`= 27/125`

 

b.  `1-text(Pr)\ (S^{′} S^{′} S^{′})`

`= 1-27/125`

`= 98/125`

 

c.  `text(Pr)\ (S^{′} S) + text(Pr)\ (S^{′} S^{′} S)`

`= 3/5 xx 2/5 + (3/5)^2 xx 2/5`

`= 48/125`

Probability, MET2 2015 VCAA 12 MC

A box contains five red balls and three blue balls. John selects three balls from the box, without replacing them.

The probability that at least one of the balls that John selected is red is

  1. `5/7`
  2. `5/14`
  3. `7/28`
  4. `15/56`
  5. `55/56`
Show Answers Only

`E`

Show Worked Solution
`text{Pr(at least 1 red)}` `= 1 – text{Pr(no reds)}`
  `= 1 – text(Pr)(B,B,B)`
  `= 1 – (3/8 xx 2/7 xx 1/6)`
  `= 55/56`

 
`=>   E`

Probability, MET2 2014 VCAA 22 MC

John and Rebecca are playing darts. The result of each of their throws is independent of the result of any other throw. The probability that John hits the bullseye with a single throw is `1/4.` The probability that Rebecca hits the bullseye with a single throw is `1/2.` John has four throws and Rebecca has two throws.

The ratio of the probability of Rebecca hitting the bullseye at least once to the probability of John hitting the bullseye at least once is

  1. `1:1`
  2. `32:27`
  3. `64:85`
  4. `2:1`
  5. `192:175`
Show Answers Only

`E`

Show Worked Solution

`X = #\ text(times John hits bullseye.)`

♦ Mean mark 37%.

`Y = #\ text(times Rebecca hits bullseye.)`

`text(Required ratio:)`

`text(Pr)(Y >= 1)` `\ :\ text(Pr)(X >= 1)`
`1-text(Pr)(Y=0)` `\ :\ 1-text(Pr)(X=0)`
`1-(1/2)^2` `\ :\ 1-(3/4)^4`
`3/4` `\ :\ 175/256`
`192/256` `\ :\ 175/256`
`:. 192` `\ :\ 175`

 
`=>   E`

Probability, MET2 2012 VCAA 20 MC

A discrete random variable  `X`  has the probability function  `text(Pr)(X = k) = (1 -p)^k p`, where `k` is a non-negative integer.

`text(Pr)(X > 1)`  is equal to

  1. `1 - p + p^2`
  2. `1 - p^2`
  3. `p - p^2`
  4. `2p - p^2`
  5. `(1 - p)^2`
Show Answers Only

`E`

Show Worked Solution
♦♦♦ Mean mark 19%.
`text(Pr)(X > 1)` `= 1 – text(Pr)(X = 0) – text(Pr)(X = 1)`
  `= 1 – [(1 – p)^0p] – [(1 – p)p]`
  `= 1 – p – (p – p^2)`
  `= 1 – 2p + p^2`
  `= (1 – p)^2`

`=>   E`