SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Probability, SM-Bank 091

In an experiment, a 6-sided die is thrown a number of times and the results are listed below.
 

\(4, 1, 3, 2, 6, 5, 2, 2, 1, 4\)

\(3, 2, 1, 1, 2, 4, 2, 6, 1, 2\)
 

  1. What is the relative frequency of getting a 2? Give your answer as a decimal.  (1 mark)
  2. If this experiment was repeated 160 times, how many times would you expect to throw a 2?  (1 mark)
Show Answers Only

a.    \(0.35\)

b.    \(56\)

Show Worked Solution

a.    \(\text{Relative frequency}=\dfrac{\text{number of times event occurs}}{\text{total number of trials}}\)

\(P(2)\) \(=\dfrac{7}{20}\)
  \(=0.35\)

 

b.     \(\text{Number of 2’s}\) \(=160\times\dfrac{7}{20}\)
    \(=56\)

\(\therefore\ \text{You would expect to throw 56 2’s in 160 trials}\)

Probability, SM-Bank 090

A coin is tossed and the results are as follows, where \(H=\)heads and \(T=\)Tails.

\(H, T, H, T, H, T, H, T, T, T, H, T, T, T\)

  1. What is the relative frequency of getting a Head?  (1 mark)
  2. If the trial was repeated 42 times, how many times would you expect to toss a head?  (1 mark)
Show Answers Only

a.    \(\dfrac{5}{14}\)

b.    \(15\)

Show Worked Solution

a.    \(\text{Relative frequency}=\dfrac{\text{number of times event occurs}}{\text{total number of trials}}\)

\(P(H)=\dfrac{5}{14}\)

b.     \(\text{Number of heads}\) \(=42\times\dfrac{5}{14}\)
    \(=15\)

\(\therefore\ \text{You would expect to toss 15 heads in 42 trials}\)

Probability, SM-Bank 089

The fish in a pond are to be removed for the pond to be cleaned. There are 4 different types of coloured carp in the pond.

Calculate the probability that the first carp captured is orange, given the probability of capturing:

  • a red carp is \(\dfrac{2}{9}\)
  • a white carp is \(\dfrac{1}{6}\) and
  • a mulit-coloured carp is \(\dfrac{5}{12}\).  (2 marks)
Show Answers Only

\(\dfrac{7}{36}\)

Show Worked Solution

\(\text{The sum of all probabilities in an event = 1}\)

\(P\text{(Orange)}\) \(=1-\Bigg(\dfrac{2}{9}+\dfrac{1}{6}+\dfrac{5}{12}\Bigg)\)
  \(=1-\Bigg(\dfrac{8}{36}+\dfrac{6}{36}+\dfrac{15}{36}\Bigg)\)
  \(=1-\dfrac{29}{36}=\dfrac{7}{36}\)

 

Probability, SM-Bank 088

A spinner with equal divisions has the numbers 1, 2 and 3 printed on it.

The probability of spinning the number 2 is \(\dfrac{1}{5}\) and the probability of spinning the number 3 is \(\dfrac{2}{3}\).

What is the probability of spinning the number 1?  (2 marks)

Show Answers Only

\(\dfrac{2}{15}\)

Show Worked Solution

\(\text{The sum of all probabilities in an event = 1}\)

\(P\text{(1)}\) \(=1-\Bigg(\dfrac{1}{5}+\dfrac{2}{3}\Bigg)\)
  \(=1-\Bigg(\dfrac{3}{15}+\dfrac{10}{15}\Bigg)\)
  \(=1-\dfrac{13}{15}=\dfrac{2}{15}\)

 

Probability, SM-Bank 087

A bag contains a number of identical red, blue and green discs.

The probability of drawing a red disc is \(\dfrac{2}{3}\) and the probability of drawing a blue disc is \(\dfrac{1}{4}\).

What is the probability of drawing a green disc?  (2 marks)

Show Answers Only

\(\dfrac{1}{12}\)

Show Worked Solution

\(\text{The sum of all probabilities in an event = 1}\)

\(P\text{(Green)}\) \(=1-\Bigg(\dfrac{2}{3}+\dfrac{1}{4}\Bigg)\)
  \(=1-\Bigg(\dfrac{8}{12}+\dfrac{3}{12}\Bigg)\)
  \(=1-\dfrac{11}{12}=\dfrac{1}{12}\)

 

Probability, SM-Bank 086

A 12 sided die is rolled. List the favourable outcomes for this event where the number rolled is less than 6.  (1 mark)

Show Answers Only

\(\text{{1, 2, 3, 4, 5}}\)

Show Worked Solution

\(\text{Possible outcomes = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}}\)

\(\text{Favourable outcomes = {1, 2, 3, 4, 5}  (Note: 6 not included)}\)

Probability, SM-Bank 085

Jordan and Degas play a board game with the spinner shown.
 

 
Jordan spins the arrow.

  1. What is the probability that Jordan spins an even number?  (1 mark)
  2. To win the game you must spin an odd number.
    What is the probability that Jordan will win the game on the next spin?  (1 mark)

Show Answers Only

a.    \(\dfrac{1}{3}\)

b.    \(\dfrac{2}{3}\)

Show Worked Solution
a.    \(P\text{(Even)}\) \(=\dfrac{\text{# even numbers}}{\text{Total numbers}}\)
    \(=\dfrac{3}{9}\)
    \(=\dfrac{1}{3}\)

 

b.    \(P\text{(Odd)}\) \(=1-P\text{(Even)}\)
    \(=1-\dfrac{1}{3}\)
    \(=\dfrac{2}{3}\)

Probability, SM-Bank 084 MC

Which percentage description best describes an event that is likely to occur?

  1. \(\text{Equal to }50\%\)
  2. \(\text{Equal to }0\%\)
  3. \(\text{Greater than }0\%\ \text{but less than }50\%\)
  4. \(\text{Greater than }50\%\ \text{but less than }100\%\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{An event that is likely to occur has a greater than }50\%\)

\(\text{but less than }100\%\ \text{chance of occurring.}\)

\(\Rightarrow D\)

Probability, SM-Bank 083

The letters from the word KITCHEN are written on cards and one is drawn at random.

  1. List the elements of the sample space.  (1 mark)
  2. List the favourable outcomes if the letter chosen is NOT a vowel (a,e, i, o, u).  (1 mark)
  3. For the letter chosen, find \(P\)(letter is a vowel).  (1 mark)
Show Answers Only

a.    \(\text{{K, I, T, C, H, E, N}}\)

b.    \(\text{{K, T, C, H, N}}\)

c.    \(\dfrac{2}{7}\)

Show Worked Solution

a.    \(\text{{K, I, T, C, H, E, N}}\)

b.    \(\text{{K, T, C, H, N}}\)

c.    \(P\text{(is a vowel)}\)

\(=\dfrac{2}{7}\)

Probability, SM-Bank 082

A spinner has 8 equal divisions numbered using the even numbers from 2 to 16.

List the sample space for this spinner.  (1 mark)

Show Answers Only

\(\text{{2, 4, 6, 8, 10, 12, 14, 16}}\)

Show Worked Solution

\(\text{Sample space = {2, 4, 6, 8, 10, 12, 14, 16}}\)

Probability, SM-Bank 081

Guy is going to throw a ten sided dice labelled with the numbers 1 to 10.

He is interested in the likelihood of various events.

  1. List the sample space for throwing this dice.  (1 mark)
  2. List the favourable outcomes if Guy wants a number greater than 7.  (1 mark)
  3. List the favourable outcomes if Guy wants an odd number less than 5.  (1 mark)
  4. Which of the events from (b) and (c) above is more likely?  (1 mark)
Show Answers Only

a.    \(\text{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}\)

b.    \(\text{{8, 9, 10}}\)

c.    \(\text{{1, 3}}\)

d.    \(\text{(b) a number > 8}\)

Show Worked Solution

a.    \(\text{Sample space = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}\)

b.    \(\text{Numbers}>7= \text{{8, 9, 10}}\)

c.    \(\text{Odd numbers}<5= \text{{1, 3}}\)

d.    \(\text{(b) a number} > 8\ \text{is more likely as it }\)

\(\text{has 1 more favourable outcome than (c).}\)

Probability, SM-Bank 080

Ahmed rolled a standard 6-sided dice a number of times. He noticed after numerous rolls that the 5 and 6 were twice as likely to occur than any of the other numbers.

Use the numbers 1, 2, 3, 4, 5 and 6 on the spinner below to create a spinner that would give the same results.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{5 and 6 twice as likely}\)

\(\longrightarrow\ \text{Spinner has 6 + 2 sections}\)

\(\therefore\ \text{5 and 6 take up 2 sections each and 1, 2, 3 and 4 one section each.}\)

Probability, SM-Bank 079

Using the numbers 1, 2 and 3 label, the spinner below so that 3 is twice as likely to occur as either of the other numbers.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{3 twice as likely}\)

\(\therefore\ \text{3 takes up 2 sections and 1 and 2 each take up 1 section.}\)

Probability, SM-Bank 078

Shade the spinner below so that white is 2 times more likely to occur than grey.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{White 2 times more likely}\)

\(\therefore\ \text{White takes up }\dfrac{8}{12}=\dfrac{2}{3}\text{ and grey}\dfrac{4}{12}=\dfrac{1}{3}.\)

Probability, SM-Bank 077

Shade the spinner below so that white is 2 times more likely to occur than grey.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{White 2 times more likely}\)

\(\therefore\ \text{White takes up }\dfrac{4}{6}=\dfrac{2}{3}\text{ and grey}\dfrac{2}{6}=\dfrac{1}{3}.\)

Probability, SM-Bank 076

Shade the spinner below so that white is 3 times more likely to occur than grey.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{White 3 times more likely}\)

\(\therefore\ \text{White each take up }\dfrac{3}{4}\text{ and grey}\dfrac{1}{4}.\)

Probability, SM-Bank 074

Given that white and yellow are equally likely, shade the spinner below so that blue is twice as likely to occur as either white or yellow.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{Blue is is twice as likely as white or yellow}\)

\(\therefore\ \text{Blue takes up half the circle and white }\)

\(\ \ \ \ \text{and yellow each take up one quarter.}\)

Probability, SM-Bank 072

Shade the spinner below so that white is 3 times more likely to occur than blue.  (1 mark)
 

Show Answers Only

Show Worked Solution

\(\text{White 3 times more likely }\)

\(\therefore\ \text{white takes up }\dfrac{3}{4}\text{ of circle and blue }\dfrac{1}{4}\)

Probability, SM-Bank 070

Tom has a dice and rolls it repeatedly 54 times, each time recording which side faces up.

How many times should he expect to see the side four coming up?  (2 marks)

Show Answers Only

\(9\text{ times}\)

Show Worked Solution
\(\text{Number of times a 4}\) \(=P\text{(4 on die)}\times 54\)
  \(=\dfrac{1}{6}\times 54\)
  \(=9\)

 
\(\therefore\ \text{A 4 would be expected 9 times.}\)

Probability, SM-Bank 069

Wendy has a number of different types of flowers as shown in the table below. 
 

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Type of Flower} \rule[-1ex]{0pt}{0pt} & \textbf{Number of Flowers} \\
\hline
\rule{0pt}{2.5ex} \text{Carnation} \rule[-1ex]{0pt}{0pt} & 8 \\
\hline
\rule{0pt}{2.5ex} \text{Tulip} \rule[-1ex]{0pt}{0pt} & 6 \\
\hline
\rule{0pt}{2.5ex} \text{Dandelion} \rule[-1ex]{0pt}{0pt} & 10 \\
\hline
\rule{0pt}{2.5ex} \text{Rose} \rule[-1ex]{0pt}{0pt} & 8 \\
\hline
\end{array}

  1. What is the probability that a randomly selected flower will be either a carnation or a tulip?  (2 marks)
  2. What is the probability that a randomly selected flower will not be a rose?  (2 marks)

Show Answers Only

a.    \(\dfrac{7}{16}\)

b.    \(\dfrac{3}{4}\)

Show Worked Solution
a.    \(P\text{(carnation or tulip)}\) \(=\dfrac{\text{number carnations + number tulips}}{\text{total number of flowers}}\)
    \(=\dfrac{8+6}{32}\)
    \(=\dfrac{14}{32}=\dfrac{7}{16}\)

 

b.    \(P\text{(not a rose)}\) \(=1-P\text{(is a rose)}\)
    \(=1-\dfrac{\text{number roses}}{\text{total number of flowers}}\)
    \(=1-\dfrac{8}{32}\)
    \(=1-\dfrac{1}{4}\)
    \(=\dfrac{3}{4}\)

Probability, SM-Bank 068 MC

A biased die has 6 faces numbered from 1 to 6.

Jackson throws the die 60 times and records the results in the table below.
 

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Number} \rule[-1ex]{0pt}{0pt} & \ \ 1\ \ & \ \ 2 \ \ & \ \ 3 \ \ & \ \ 4 \ \ & \ \ 5 \ \ & \ \ 6 \ \ \\
\hline
\rule{0pt}{2.5ex} \textbf{Times} \rule[-1ex]{0pt}{0pt} & \ \ 8\ \ & \ \ 14 \ \ & \ \ 9 \ \ & \ \ 13 \ \ & \ \ 7 \ \ & \ \ 9 \\
\hline
\end{array} 

Using the table, what is the probability that Jackson throws a 2 on his next throw?

  1. \(\dfrac{7}{30}\)
  2. \(\dfrac{14}{46}\)
  3. \(\dfrac{1}{5}\)
  4. \(\dfrac{1}{6}\)

Show Answers Only

\(A\)

Show Worked Solution
\(P(2)\) \(=\dfrac{\text{Number of 2’s}}{\text{Total number of throws}}\)
  \(=\dfrac{14}{60}\)
  \(=\dfrac{7}{30}\)

 
\(\Rightarrow A\)

Probability, SM-Bank 067

A bag of coloured balls contains 10 blue balls, 12 white balls, 5 black balls, 2 red balls, and 9 orange balls.

Joyce grabs a ball from the bag without looking what is inside.

What is the probability that she grabs an white ball?  (1 mark)

Show Answers Only

\(\dfrac{6}{19}\)

Show Worked Solution
\(P\text{(White ball)}\) \(=\dfrac{\text{Total white balls}}{\text{Total balls}}\)
  \(=\dfrac{12}{10+12+5+2+9}\)
  \(=\dfrac{12}{38}\)
  \(=\dfrac{6}{19}\)

Probability, SM-Bank 066

A bag contains different coloured balls.

It contains 4 red balls, 8 green balls, 3 pink balls, 6 grey balls and 11 yellow balls.

Nathalie picks a ball from the bag without looking.

  1. What is the chance that she draws a grey ball?  (1 mark)
  2. What is the chance that the ball she draws in not pink?  (2 marks)

Show Answers Only

a.    \(\dfrac{3}{16}\)

b.    \(\dfrac{29}{32}\)

Show Worked Solution
a.    \(P\text{(Grey ball)}\) \(=\dfrac{\text{Number of grey balls}}{\text{Total number of balls}}\)
    \(=\dfrac{6}{4+8+3+6+11}\)
    \(=\dfrac{6}{32}\)
    \(=\dfrac{3}{16}\)

 

b.    \(P\text{(not pink ball)}\) \(=1-P\text{(pink ball)}\)
    \(=1-\dfrac{\text{Number of pink balls}}{\text{Total number of balls}}\)
    \(=1-\dfrac{3}{32}\)
    \(=\dfrac{29}{32}\)

Probability, SM-Bank 064

A biased die is rolled.

The probability of rolling an even number is 56%.

What is the probability of rolling an odd number?  (1 mark)

Show Answers Only

\(44\%\)

Show Worked Solution

\(P\text{(odd number)}\)

  \(=1-P\text{(even number)}\)
  \(=1-0.56\)
  \(=0.44\)
  \(=44\%\)

Probability, SM-Bank 063

Victoria runs a pet store that sells pure breed dogs.

She has several dogs in the shop from different breeds.
 


 

  1. If one of Victoria's dogs is chosen at random, what is the probability it will be either a labrador or a pug?  (1 mark)

  2. If one of the dogs is chosen at random, what is the probability that the dog is not a bulldog?  (2 marks)
Show Answers Only

a.    \(\dfrac{5}{9}\)

b.    \(\dfrac{5}{6}\)

Show Worked Solution
a.    \(P\text{(Labrador or Pug)}\) \(=\dfrac{\text{Number of Labrador and Pug}}{\text{Total dogs}}\)
    \(=\dfrac{16 + 4}{4+16+6+5+5}\)
    \(=\dfrac{20}{36}=\dfrac{5}{9}\)

 

b.    \(P\text{(Dog not a bulldog)}\) \(=1-P\text{(Dog is a bulldog)}\)
    \(=1-\dfrac{6}{36}\)
    \(=\dfrac{30}{36}=\dfrac{5}{6}\)

Probability, SM-Bank 062

Tristan's laundromat has a lost clothing basket that contains only black and white socks.

The probability of randomly picking a black sock from the basket is 35%.

What is the probability of randomly picking a white sock?  (1 mark)

Show Answers Only

\(65\%\)

Show Worked Solution

\(P\text{(white)}+P\text{(Black)}=100\%\)

\(\therefore\ P\text{(white)}\) \(=100-P\text{(black)}\)
  \(=100-35\)
  \(=65\%\)

Probability, SM-Bank 061

Chusi uses this net to make a dice.
 

  
  1. Chusi rolls the dice once. What is the chance that Chusi will roll a 2? (1 mark)
  2. Chusi makes up a game and to win the game you must roll a number larger than 2.
    What is the chance that Chusi will win the game on her next roll?  (1 mark)

  3. If Chusi rolls the dice 108 times, how many times could she expect to win the game?  (1 mark)
Show Answers Only

a.    \(\dfrac{1}{3}\)

b.    \(\dfrac{1}{6}\)

c.    \(18\ \text{times}\)

Show Worked Solution
a.    \(P(2)\) \(=\dfrac{\text{number of 2’s}}{\text{total possibilities}}\)
    \(=\dfrac{2}{6}\)
    \(=\dfrac{1}{3}\)

 

b.    \(P\text{(number>2)}\) \(=\dfrac{\text{number of 3’s}}{\text{total possibilities}}\)
    \(=\dfrac{1}{6}\)

\(\therefore\ \text{Chusi’s chance of winning on the next roll is }\dfrac{1}{6}.\)
 

c.    \(\text{Expected wins}\) \(=\text{number of rolls}\times \text{probability of winning}\)
    \(=108\times \dfrac{1}{6}\)
    \(=18\)

\(\therefore\ \text{Chusi could expect to win 18 times with 108 rolls.}\)

Probability, SM-Bank 060 MC

Benji randomly picks one disc from the group of white and coloured discs, pictured below.
 

 

 
What is the probability Benji chooses a white disc?

  1. \(\dfrac{2}{3}\)
  2. \(\dfrac{8}{13}\)
  3. \(\dfrac{8}{21}\)
  4. \(\dfrac{13}{21}\)
Show Answers Only

\(D\)

Show Worked Solution
\(P\text{(White disc)}\) \(=\dfrac{\text{Number of white discs}}{\text{Total discs}}\)
  \(=\dfrac{13}{21}\)

\(\Rightarrow D\)

Probability, SM-Bank 059 MC

Ashanti works in the city on weekdays and has weekends off.

She catches a bus to work 3 days each week and drives two days.

If today is a weekday, what is the probability that Ashanti catches a bus to work?

  1. \(3\%\)
  2. \(0.30\)
  3. \(\dfrac{3}{7}\)
  4. \(\dfrac{3}{5}\)
Show Answers Only

\(D\)

Show Worked Solution
\(P\text{(Bus)}\) \(=\dfrac{\text{favourable events}}{\text{total possible events}}\)
  \(=\dfrac{3}{5}\)

\(\Rightarrow D\)

Probability, SM-Bank 058 MC

A bag of balls contains only blue and white balls.

Jimoen picks one ball from the bag.

What is the chance the ball is white?

  1. \(\dfrac{2}{7}\)
  2. \(\dfrac{1}{3}\)
  3. \(\dfrac{2}{5}\)
  4. \(\dfrac{5}{7}\)
Show Answers Only

\(D\)

Show Worked Solution
\(P\text{(white)}\) \(=\dfrac{\text{Number of white balls}}{\text{Total number of balls}}\)
  \(=\dfrac{5}{7}\)

\(\Rightarrow D\)