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Probability
NOTE: Clicking an entry in the right column will take you directly to that question!
(BLUE = Paper 1, RED = Paper 2)
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Principle of Counting |
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Arrangements |
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Sample Spaces |
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Tree Diagrams |
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Venn Diagrams |
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Expected Values |
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Addition Rule |
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Multiplication Rule |
2022 Paper 2 Question 3(b)
(b) Ben has to choose three subjects to study.
He must pick one subject from each of these three groups:
| Group A | Group B | Group C |
|---|---|---|
|
French Spanish German |
Biology Physics Chemistry Business |
Art Accounting History Geography Home Economics |
(i) How many different choices of three subjects can Ben make, picking one from each group?
(ii) The school is going to add one extra subject to one of these groups (A, B, or C).
Which group should the extra subject be added to, in order to make the number of different choices of three subjects that Ben can make as large as possible?
Justify your answer.
(i) \(60\)
(ii) Group A
(i)
\begin{align}3\times4\times5=60\end{align}
(ii)
\begin{align}\mathbf{A}:4\times4\times5=80\end{align}
or
\begin{align}\mathbf{B}:3\times5\times5=75\end{align}
or
\begin{align}\mathbf{C}:3\times4\times6=72\end{align}
It should therefore be added to Group A.
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2022 Paper 2 Question 7(b)-(e)
The table below shows the breakdown of animals at a shelter, by type of animal (cat or dog) and by sex (male or female), on a particular Monday.
(b) Complete the table, by filling in the four missing values.
| Male | Female | Total | |
|---|---|---|---|
|
Cats |
\(5\) |
\(9\) |
\(\mathbf{14}\) |
|
Dogs |
\(11\) |
|
|
|
Total |
|
|
\(\mathbf{40}\) |
| Male | Female | Total | |
|---|---|---|---|
|
Cats |
\(5\) |
\(9\) |
\(\mathbf{14}\) |
|
Dogs |
\(11\) |
\(15\) |
\(\mathbf{26}\) |
|
Total |
\(16\) |
\(24\) |
\(\mathbf{40}\) |
| Male | Female | Total | |
|---|---|---|---|
|
Cats |
\(5\) |
\(9\) |
\(\mathbf{14}\) |
|
Dogs |
\(11\) |
\(15\) |
\(\mathbf{26}\) |
|
Total |
\(16\) |
\(24\) |
\(\mathbf{40}\) |
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Three different animals were picked at random from the animals in the shelter on this Monday.
(c)
(i) Find the probability that the first animal picked was a cat.
(ii) Find the probability that all three animals picked were male dogs.
Give your answer correct to \(3\) decimal places.
(i) \(\dfrac{7}{20}\)
(ii) \(0.017\)
(i)
\begin{align}P(\mbox{cat})&=\frac{14}{40}\\&=\frac{7}{20}\end{align}
(ii)
\begin{align}P(3\mbox{ male dogs})&=\frac{11}{40}\times\frac{10}{39}\times\frac{9}{38}\\&\approx0.017\end{align}
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(d) The \(9\) female cats were put in \(9\) separate pens.
Work out the number of ways in which this could have been done (that is, the number of different possible arrangements).
\(362{,}880\)
\begin{align}9!=362{,}880\end{align}
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(e) By the end of this week, \(10\) of the animals had left the shelter, and no new animals had been taken in. If an animal was picked at random at the end of this week, the probability of picking a dog would be \(\dfrac{11}{15}\).
Work out how many cats left the shelter during this week.
\(6\)
\begin{align}\mbox{Number of dogs}&=\frac{11}{15}\times(40-10)\\&=22\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\mbox{Number of cats}&=30-22\\&=8\end{align}
Therefore, the number of cats that left is \(14-8=6\).
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2021 Paper 2 Question 1
The game “Twister” is played using two spinners and a mat with coloured spots.
The two spinners needed for the game are shown below. Both consist of four equal sections.
A player spins both spinners and must then place their selected hand/foot on the selected
coloured spot on the game mat.
The spinners below show the outcome Right Hand on a Yellow Spot.
(a) How many different possible outcomes are there in the game?
\(16\)
\begin{align}4\times4=16\end{align}
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(b) Kate spins each spinner once. What is the probability that she must use:
(i) her left foot?
(ii) a red or yellow coloured spot?
(i) \(\dfrac{1}{4}\)
(ii) \(\dfrac{1}{2}\)
(i)
\begin{align}\frac{1}{4}\end{align}
(ii)
\begin{align}\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\end{align}
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(c) Jack spins each spinner once. What is the probability that the outcome is:
(i) his right hand and a blue spot?
(ii) his right hand or a blue spot?
(i) \(\dfrac{1}{16}\)
(ii) \(\dfrac{7}{16}\)
(i)
\begin{align}\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\end{align}
(ii)
\begin{align}\frac{1}{4}+\frac{1}{4}-\frac{1}{16}=\frac{7}{16}\end{align}
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2021 Paper 2 Question 2(a)
(a) At a stall in a fun fair, the probability of knocking a coconut off its support is \(0.34\).
(i) What is the probability of not knocking a coconut off its support?
(ii) David is given three attempts to knock the coconut off its support.
What is the probability that he knocks the coconut for the first time, on his third attempt? Give your answer correct to three decimal places.
(i) \(0.66\)
(ii) \(0.148\)
(i)
\begin{align}P&=1-0.34\\&=0.66\end{align}
(ii)
\begin{align}P&=0.66\times0.66\times0.34\\&\approx0.148\end{align}
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2020 Paper 2 Question 1
(a) A restaurant is offering a three-course meal consisting of one starter, one main course and
one dessert. There are \(4\) different starters, \(6\) different main courses and \(8\) different desserts
to choose from.
(i) How many different three-course meal combinations are available?
(ii) When Jack visits the restaurant he discovers that the restaurant still has the \(4\) starters and the \(6\) main courses available but is sold out of some of the desserts.
Jack now has \(120\) different three-course meal combinations to choose from.
How many different desserts are still available to Jack?
(i) \(192\)
(ii) \(5\)
(i)
\begin{align}4\times6\times8=192\end{align}
(ii)
\begin{align}4\times 6\times x=120\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{120}{4\times6}\\&=5\end{align}
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(b) In a large population \(1\) in \(8\) of the people play tennis.
(i) Four people are chosen at random from the population.
What is the probability that the fourth person chosen is the only one to play tennis?
(ii) Three people are chosen at random from the population.
What is the probability that exactly two of them play tennis?
(i) \(\dfrac{343}{4096}\)
(ii) \(\dfrac{21}{512}\)
(i)
\begin{align}\frac{7}{8}\times\frac{7}{8}\times\frac{7}{8}\times\frac{1}{8}=\frac{343}{4096}\end{align}
(ii)
\begin{align}{3\choose2}\times\left(\frac{1}{8}\right)^2\times\frac{7}{8}=\frac{21}{512}\end{align}
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2019 Paper 2 Question 1(c)
A business has \(28\) employees.
Their ages, in years, are given below.
\begin{align}32&&41&&57&&64&&19&&21&&35\\ 18&&43&&54&&63&&65&&33&&22\\ 39&&58&&18&&42&&20&&34&&21\\ 49&&33&&55&&34&&57&&43&&63\end{align}
This information is represented on the following stem-and-leaf diagram.
(c) One employee is chosen at random on a day when all employees are present at work.
(i) Find the probability that the employee is a teenager (\(<20\) years of age).
(ii) Find the probability that the employee chosen is a person in their thirties whose age is even or a person in their forties whose age is odd.
(i) \(\dfrac{3}{28}\)
(ii) \(\dfrac{7}{28}\)
(i)
\begin{align}\frac{3}{28}\end{align}
(ii)
\begin{align}\frac{3}{28}+\frac{4}{28}=\frac{7}{28}\end{align}
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2019 Paper 2 Question 3
In a population, the probability that a person has blue eyes is \(0.7\).
(a) One person is chosen at random from the population.
What is the probability that this person does not have blue eyes?
\(0.3\)
\begin{align}P&=1-0.7\\&=0.3\end{align}
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(b) Two people are chosen at random.
What is the probability that both have blue eyes?
\(0.49\)
\begin{align}P&=0.7\times0.7\\&=0.49\end{align}
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(c) Three people are chosen at random.
What is the probability that exactly two of them have blue eyes?
\(0.441\)
\begin{align}P&={3\choose2}\times0.7^2\times0.3^1\\&=0.441\end{align}
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(d) Four people are chosen at random, one after another.
What is the probability that the fourth person of the four chosen is the only one to have blue
eyes?
\(0.0189\)
\begin{align}P&=0.3^3\times0.7\\&=0.0189\end{align}
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2018 Paper 2 Question 1
(a) An experiment consists of throwing two fair, standard, six-sided dice and recording the sum of the two numbers thrown. Some of the totals are shown in the table.
(a) An experiment consists of throwing two fair, standard, six-sided dice and recording the sum of the two numbers thrown. Some of the totals are shown in the table.
(i) Complete the table.
(ii) Find the probability of getting a total of \(7\) or \(11\).
(iii) Find the probability of getting a total which is a prime number.
(i)
(ii) \(\dfrac{2}{9}\)
(iii) \(\dfrac{5}{12}\)
(i)
(ii)
\begin{align}P&=\frac{6+2}{36}\\&=\frac{2}{9}\end{align}
(iii)
\begin{align}P&=\frac{1+2+4+6+2}{36}\\&=\frac{5}{12}\end{align}
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(b) A car distributor sells Ford cars and Renault cars.
It has \(30\) cars for sale on a particular day; \(18\) are Ford cars and \(12\) are Renault cars.
\(7\) of the Ford cars are red and \(4\) of the Renault cars are red. One of the \(30\) cars is chosen at random. What is the probability that the car chosen is a Ford car or a car which is not red?
\(\dfrac{13}{15}\)
\begin{align}P&=\frac{18}{30}+\frac{8}{30}\\&=\frac{13}{15}\end{align}
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2018 Paper 2 Question 3
(a)
(i) Find the number of different arrangements that can be made using all the letters of the word RAINBOW. Each letter is used only once.
(ii) Find the number of different \(3\)-letter arrangements that can be made using the letters
of the word RAINBOW. Each letter is used at most once.
(i) \(5{,}040\)
(ii) \(210\)
(i)
\begin{align}7!=5{,}040\end{align}
(ii)
\begin{align}7\times6\times5=210\end{align}
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(b) A game, called Rainbow, uses an unbiased circular spinner.
The spinner has seven sectors coloured red (\(R\)), orange (\(O\)), yellow (\(Y\)), green (\(G\)), blue (\(B\)), indigo (\(I\)), and violet (\(V\)) as shown below.
The table below shows the angle of each sector.
It also shows the cash prize that a player wins if the spinner stops in that sector.
| Colour | Angle | Probability | Prize |
|---|---|---|---|
|
Red |
\(72^{\circ}\) |
|
\(20\mbox{ euro}\) |
|
Orange |
\(30^{\circ}\) |
|
\(60\mbox{ euro}\) |
|
Yellow |
\(45^{\circ}\) |
\(\dfrac{1}{8}\) |
\(24\mbox{ euro}\) |
|
Green |
\(90^{\circ}\) |
|
\(8\mbox{ euro}\) |
|
Blue |
\(60^{\circ}\) |
|
\(42\mbox{ euro}\) |
|
Indigo |
\(18^{\circ}\) |
|
\(90\mbox{ euro}\) |
|
Violet |
\(45^{\circ}\) |
|
\(48\mbox{ euro}\) |
(i) Complete the “Probability” column of the table which shows the probability of the spinner coming to rest in each sector after one spin.
(ii) Find the expected value of the prize that a player wins if they play Rainbow.
(i)
| \(t\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|---|
|
\(\mathbf{h(t)}\) |
|
\(42\) |
|
|
|
|
|
|
|
(ii) \(31.50\mbox{ euro}\)
(i)
| \(t\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|---|
|
\(\mathbf{h(t)}\) |
|
\(42\) |
|
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(ii)
\begin{align}E(X)&=20\left(\frac{1}{5}\right)+60\left(\frac{1}{12}\right)+24\left(\frac{1}{8}\right)+8\left(\frac{1}{4}\right)+42\left(\frac{1}{6}\right)+90\left(\frac{1}{20}\right)+48\left(\frac{1}{8}\right)\\&=31.50\mbox{ euro}\end{align}
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2017 Paper 2 Question 1
According to the Central Statistics Office (CSO) there were \(65{,}909\) babies born in Ireland in \(2015\).
Of these \(32{,}290\) were girls.
(a)
(i) How many boys were born in Ireland in \(2015\)?
(ii) Find the probability that a baby picked at random from those born in Ireland in \(2015\) is a boy. Give your answer correct to \(2\) decimal places.
(i) \(33{,}619\)
(ii) \(0.51\)
(i)
\begin{align}65{,}909-32{,}290=33{,}619\end{align}
(ii)
\begin{align}P&=\frac{33{,}619}{65{,}909}\\&\approx0.51\end{align}
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(b) Eight babies were born in Limerick’s Maternity Hospital on \(1\) May \(2015\).
(i) Use your answer to part (a)(ii) to find the probability that the first three babies born were boys. Give your answer correct to \(4\) decimal places.
(ii) Find the probability that the third birth was the first girl born in the hospital that day.
Give your answer correct to \(4\) decimal places.
(i) \(0.1327\)
(ii) \(0.1274\)
(i)
\begin{align}P&=0.51^3\\&\approx0.1327\end{align}
(ii)
\begin{align}P&=0.51^2\times0.49\\&\approx0.1274\end{align}
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(c) The table below shows the probability of being born on a particular day of the week.
(i) Complete the table.
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
|
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii) In a particular week \(1300\) babies were born.
Find the number of babies expected to be born on the Tuesday of that week.
(i)
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
\(0.16\) |
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii) \(208\)
(i)
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
|
Probability |
\(0.14\) |
\(0.16\) |
\(0.15\) |
\(0.18\) |
\(0.15\) |
\(0.12\) |
\(0.1\) |
(ii)
\begin{align}1{,}300\times0.16=208\end{align}
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2017 Paper 2 Question 5(b)
(b) A class of \(24\) pupils is made up of \(10\) boys and \(14\) girls. Chemistry is studied by \(6\) of the boys and \(9\) of the girls. A pupil is chosen at random from the class. Find the probability that the pupil chosen is a boy or is a pupil who does not study chemistry.
\(\dfrac{15}{24}\)
\begin{align}P&=\frac{10}{24}+\frac{5}{24}\\&=\frac{15}{24}\end{align}
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2016 Paper 2 Question 1
(a) A survey of \(168\) people was carried out. Participants were asked whether they owned a cat or a dog. Some of the results are recorded in the Venn diagram below.
(i) Of those surveyed, \(19\) people did not own either a cat or a dog. Complete the diagram.
(i) Of those surveyed, \(19\) people did not own either a cat or a dog. Complete the diagram.
(i)
(ii) \(\dfrac{64}{168}\)
(iii) \(50.6\%\)
(i)
(ii)
\begin{align}\frac{64}{168}\end{align}
(iii)
\begin{align}\frac{37+48}{168}&=50.59\\&=50.6\%\end{align}
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(b) Mandy wants to buy a ticket to the theatre. She can choose a ticket in the Balcony (\(B\)) or the Stalls (\(S\)). Mandy can go on Monday (\(M\)), Wednesday (\(W\)) or Friday (\(F\)). For an extra charge she can choose a VIP ticket (\(V\)) where she will meet the band or she can choose the show only (\(O\)).
Complete the tree diagram below, and hence or otherwise, find the probability that Mandy chooses a VIP ticket on a Wednesday.
You may assume that all choices are equally likely.
\(\dfrac{1}{6}\)
\begin{align}P&=\frac{1}{12}+\frac{1}{12}\\&=\frac{1}{6}\end{align}
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2015 Paper 2 Question 1
A bank issues a unique six-digit password to each of its online customers. The password may contain any of the numbers \(0\) to \(9\) in any position and numbers may be repeated. For example, the following is a valid password.
\begin{align}\fbox{0}&&\fbox{7}&&\fbox{1}&&\fbox{7}&&\fbox{3}&&\fbox{7}\end{align}
(a) How many different passwords are possible?
\(1{,}000{,}000\)
\begin{align}10\times10\times10\times10\times10\times10=1{,}000{,}000\end{align}
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(b)
(i) How many different passwords do not contain any zero?
(ii) One password is selected at random from all the possible passwords. What is the
probability that this password contains at least one zero?
(i) \(531{,}441\)
(ii) \(0.0468559\)
(i)
\begin{align}9\times9\times9\times9\times9\times9=531{,}441\end{align}
(ii)
\begin{align}P&=\frac{1{,}000{,}000-531{,}441}{1{,}000{,}000}\\&=0.0468559\end{align}
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(c) John is issued with one such password from the bank. Each time John wants to access his account online, the bank’s website requires him to input three of his password digits into the boxes provided. For example, he may be asked for the 2nd, 4th and 5th digits.
In how many different ways can the bank select the three required boxes?
\(20\)
\begin{align}{6\choose3}=20\end{align}
