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Probability, SMB-012

Two dice are rolled. What is the probability that only one of the dice shows a three?   (2 marks)

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`5/18`

Show Worked Solution

`text{Each dice roll is an independent event.}`

`P(3) = 1/6, \ P\text{(not 3)} = 1-1/6=5/6`

`text{P (Only one 3)}`

`= P text{(3, not 3)} + P text{(not 3, 3)}`

`= 1/6 xx 5/6 + 5/6 xx 1/6`

`= 10/36`

`= 5/18`

Probability, SMB-010

Each time she throws a dart, the probability that Gaga hits the dartboard is  `4/7`.

She throws two darts, one after the other.

What is the probability that she misses the dartboard with both darts?   (2 marks)

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`9/49`

Show Worked Solution

`P text{(hits)} = 4/7\ \ =>\ \ P\text{(misses)} = 1-4/7=3/7`

`P text{(misses twice)}` `= 3/7 xx 3/7`
  `= 9/49`

Probability, SMB-009

Jon spins each pointer 50 times.

 

Each time he added the numbers that the pointers landed on.

His results are shown below.

\begin{array} {|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textbf{Sum of numbers}\rule[-1ex]{0pt}{0pt} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \textbf{Total}\\
\hline
\rule{0pt}{2.5ex}\textbf{Number of spins}\rule[-1ex]{0pt}{0pt} & 1 & 2 & 3 & 6 & 8 & 9 & 7 & 5 & 4 & 4 & 1 & \textbf{50} \\
\hline
\end{array}

What percentage of the spins resulted in a sum of 9?   (2 marks)

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`text(10%)`

Show Worked Solution
`text(Percentage)` `= text(number of 9’s)/text(number of spins) xx 100`
  `=5/50 xx 100`
  `=10 text(%)`

Probability, SMB-007 MC

Brandi spins the arrow on two identical spinners.

The arrow on each spinner is equally likely to land on 1, 2 or 3.
 

If Brandi adds up the two results, which total is she least likely to get?

  1. `2`
  2. `3`
  3. `4`
  4. `5`
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`A`

Show Worked Solution

`text{2 can only result from (1, 1).}`

`text(All other totals can have more than 1 combination producing them.)`

`=>A`

Probability, SMB-005 MC

Two standard dice are rolled at the same time and the two numbers are added up.
 

nap-159-nc03 

Which total is most likely?

  1. `2`
  2. `3`
  3. `4`
  4. `6`
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`D`

Show Worked Solution

`text(A total of 6 is most likely.)`

`text{Note that a 6 can occur in the following ways:}`

`(5,1), (1,5), (4,2), (2,4) and (3,3)`

`text(No other option given has as many combinations.)`

Probability, SMB-003 MC

Arun flips an unbiased coin 200 times.

Which result is most likely?

  1. `20\ text(tails)`
  2. `98\ text(tails)`
  3. `108\ text(tails)`
  4. `196\ text(tails)`
Show Answers Only

`B`

Show Worked Solution

`text{Each toss is an independent event with 50% chance for both heads and tails}.`

`text{The expected result after 200 tosses is 100 tails, 100 heads.}`

`:.\ text{The most likely result = 98 tails (closest to 100)}`

`=> B`

Probability, SMB-002

A representative soccer team is chosen from 30 players who play for two clubs, Portland and Lithgow.

10 players from Portland and 20 players from Lithgow are playing in the trials, and 7 players from Portland and 8 from Lithgow are selected in the representative team.
 

 
One player at the trial is randomly selected.

What is the probability that the player is from Lithgow and is selected in the representative team?

Give your answer to two decimal places.   (2 marks)

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`0.27`

Show Worked Solution

`Ptext{(player is from Lithgow and selected)}`

`= P(A and B)`

`= P(A) xx P(B)\ \ \ \text{(independent events)}`

`= 20/30 xx 8/20`

`= 8/30`

`= 0.27\ \text{(2 d.p.)}`

Probability, SMB-001

A coin is tossed 3 times. There are 8 possible outcomes.

What is the probability of getting 2 heads and 1 tail in any order?   (2 marks)

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`3/8`

Show Worked Solution

`text(Strategy 1)`

`text{The table (array) below lists the possible outcomes:}`
 

  1st toss H H H H T T T T
  2nd toss H H T T H H T T
  3rd toss H T H T H T H T
       ✓  ✓    ✓      

`text(From table,)\ Ptext{(2H, 1T)} = 3/8`
  

`text(Strategy 2)`

`text(Using a probability tree:)`
 

Probability, STD2 S2 2015 HSC 16 MC

The probability of winning a game is `7/10`.

Which expression represents the probability of winning two consecutive games?

  1. `7/10 xx 6/9`
  2. `7/10 xx 6/10`
  3. `7/10 xx 7/9`
  4. `7/10 xx 7/10`
Show Answers Only

`D`

Show Worked Solution

`text{Since the two events are independent:}`

`P text{(W)}` `= 7/10`
`P text{(WW)}` `= 7/10 xx 7/10`

 
`=>D`

Probability, STD2 S2 2014 HSC 16 MC

In Mathsville, there are on average eight rainy days in October.

Which expression could be used to find a value for the probability that it will rain on two consecutive days in October in Mathsville?

  1. `8/31 xx 7/30`
  2. `8/31 xx 7/31`
  3. `8/31 xx 8/30`
  4. `8/31 xx 8/31`
Show Answers Only

`D`

Show Worked Solution

`P text{(rains)} = 8/31\ \ \text{(independent event for each day)}`

`text{Since each day has same probability:}`

`P(R_1 R_2) = 8/31 xx 8/31`

`=>  D`

♦♦♦ Mean mark 16%.
Lowest mark of any MC question in 2014!

Probability, STD2 S2 2011 HSC 15 MC

An unbiased coin is tossed 10 times.

A tail is obtained on each of the first 9 tosses.

What is the probability that a tail is obtained on the 10th toss?

  1. `1/2^10`
  2. `1/2`
  3. `1/10`
  4. `9/10`
Show Answers Only

`B`

Show Worked Solution

`text(Each toss is an independent event and has an even chance)`

`text(of being a head or tail.)`

`=> B`

Probability, STD2 S2 2013 HSC 26c

The probability that Michael will score more than 100 points in a game of bowling is `31/40`. 

  1. A commentator states that the probability that Michael will score less than 100 points in a game of bowling is  `9/40`.

     

    Is the commentator correct? Give a reason for your answer.   (1 mark)

  2. Michael plays two games of bowling. What is the probability that he scores more than 100 points in the first game and then again in the second game?   (1 mark)

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  1. `text{Incorrect. Less than “or equal to 100” is correct.}`
  2. `961/1600`
Show Worked Solution
♦♦♦ Mean mark 11%

i.   `text(The commentator is incorrect. The correct)`

`text(statement is)\ Ptext{(score} <=100 text{)} =9/40`

`text{(i.e. less than “or equal to 100” is the correct statement)}`

 

♦ Mean mark 34%
ii. `\ \ \ P(text{score >100 in both})` `= 31/40 xx 31/40` 
    `= 961/1600`

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 2013 HSC 18 MC

Two unbiased dice, each with faces numbered  1, 2, 3, 4, 5, 6,  are rolled.

What is the probability of obtaining a sum of 6?

  1. `1/6`
  2. `1/12`
  3. `5/12`
  4. `5/36`
Show Answers Only

`D`

Show Worked Solution

`text(Total outcomes)=6xx6=36`

`text{Outcomes that sum to 6}=text{(1,5) (5,1) (2,4) (4,2) (3,3)} =5`

`:.\ P\text{(sum of 6)} =5/36`

`=>\ D`

♦♦ Mean mark 35%.